Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Calculus 1
Nov 24, 2025
38
5 pages
Emily @mephdapumkin
Limits are one of the most important concepts in calculus - they help us understand what happens to... Show more
Ever wondered what happens to a function right before it hits a point where it's undefined? That's exactly what limits help us figure out! The key thing to remember is that a limit tells us about the behavior near a point, not necessarily at the point itself.
The notation means "the limit of f(x) as x approaches c equals L." Here's the cool part whether f(x) actually exists at x = c doesn't matter for the limit to exist. The function could equal something totally different at that point, or not exist there at all!
There are three main ways limits can fail to exist, and you'll see these patterns everywhere. Different end behaviors happen when approaching from left and right gives different values. Unbound behavior occurs when the function shoots off to infinity. Oscillating behavior happens when the function bounces around without settling on any value.
Quick Tip Always check what's happening on both sides of the point you're approaching - if they don't match, the limit doesn't exist!
Reading limits from graphs is like being a detective - you're looking for clues about where the function is heading. When evaluating , ignore what's actually happening at x = c and focus on the surrounding behavior.
Look at both sides of your target point. If the function approaches the same y-value from both directions, that's your limit! If the left and right sides approach different values, then the limit does not exist (DNE).
Don't get tricked by holes in the graph or jump discontinuities. A function can have a limit at a point even if there's a hole there, and the actual function value at that point might be completely different from the limit.
Graph Reading Pro Tip Use your finger to trace along the curve from both sides - where would you naturally expect to land if the hole wasn't there?
When solving limits algebraically, follow this simple three-step strategy that works every time. First, try direct substitution - just plug in the number. If it works and gives you a real number, you're done!
If direct substitution gives you something like 0/0 or makes the function undefined, then it's time for step two do more math. This usually means factoring, rationalizing, or simplifying to cancel out problematic terms.
The properties of limits make calculations way easier once you get the hang of them. You can split apart sums, differences, and products into separate limits. You can also pull out constants and handle powers naturally. Just remember that for quotients, the denominator's limit can't be zero!
Strategy Success Most limit problems that look impossible at first just need some algebraic manipulation to reveal their true nature.
When direct substitution fails, you've got some powerful algebraic tools in your toolkit. The most common situation is getting 0/0, which means you need to find a way to cancel out the problematic factors that are causing the zero.
Factoring is your best friend here - especially with polynomials. Look for common factors in the numerator and denominator that you can cancel out. For expressions with square roots, try rationalizing by multiplying by the conjugate to eliminate the radical.
Complex fractions within limits might look scary, but they follow the same principle. Find a common denominator, simplify, and then cancel what you can. The key is transforming the expression into something where direct substitution actually works.
Algebra Reminder Every "impossible" limit problem becomes possible once you find the right algebraic technique to simplify it.
Piecewise functions are where limits get really interesting because different rules apply on different sides of a point. To find the limit at a boundary point, you need to check what happens approaching from both the left and right sides.
Calculate the left-hand limit using the piece that applies when x is less than your target value. Then find the right-hand limit using the piece that applies when x is greater than your target. If these two values match, that's your limit!
Here's the crucial point even if the piecewise function has a specific value defined at the boundary point, that doesn't affect the limit. The limit only cares about the approaching behavior, not the actual value at that spot.
Piecewise Strategy When in doubt, always check both sides separately - piecewise functions love to have different behaviors on each side!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Emily
@mephdapumkin
Limits are one of the most important concepts in calculus - they help us understand what happens to a function as we get closer and closer to a specific point, even when the function might be undefined at that exact... Show more
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ever wondered what happens to a function right before it hits a point where it's undefined? That's exactly what limits help us figure out! The key thing to remember is that a limit tells us about the behavior near a point, not necessarily at the point itself.
The notation means "the limit of f(x) as x approaches c equals L." Here's the cool part: whether f(x) actually exists at x = c doesn't matter for the limit to exist. The function could equal something totally different at that point, or not exist there at all!
There are three main ways limits can fail to exist, and you'll see these patterns everywhere. Different end behaviors happen when approaching from left and right gives different values. Unbound behavior occurs when the function shoots off to infinity. Oscillating behavior happens when the function bounces around without settling on any value.
Quick Tip: Always check what's happening on both sides of the point you're approaching - if they don't match, the limit doesn't exist!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Reading limits from graphs is like being a detective - you're looking for clues about where the function is heading. When evaluating , ignore what's actually happening at x = c and focus on the surrounding behavior.
Look at both sides of your target point. If the function approaches the same y-value from both directions, that's your limit! If the left and right sides approach different values, then the limit does not exist (DNE).
Don't get tricked by holes in the graph or jump discontinuities. A function can have a limit at a point even if there's a hole there, and the actual function value at that point might be completely different from the limit.
Graph Reading Pro Tip: Use your finger to trace along the curve from both sides - where would you naturally expect to land if the hole wasn't there?
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When solving limits algebraically, follow this simple three-step strategy that works every time. First, try direct substitution - just plug in the number. If it works and gives you a real number, you're done!
If direct substitution gives you something like 0/0 or makes the function undefined, then it's time for step two: do more math. This usually means factoring, rationalizing, or simplifying to cancel out problematic terms.
The properties of limits make calculations way easier once you get the hang of them. You can split apart sums, differences, and products into separate limits. You can also pull out constants and handle powers naturally. Just remember that for quotients, the denominator's limit can't be zero!
Strategy Success: Most limit problems that look impossible at first just need some algebraic manipulation to reveal their true nature.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When direct substitution fails, you've got some powerful algebraic tools in your toolkit. The most common situation is getting 0/0, which means you need to find a way to cancel out the problematic factors that are causing the zero.
Factoring is your best friend here - especially with polynomials. Look for common factors in the numerator and denominator that you can cancel out. For expressions with square roots, try rationalizing by multiplying by the conjugate to eliminate the radical.
Complex fractions within limits might look scary, but they follow the same principle. Find a common denominator, simplify, and then cancel what you can. The key is transforming the expression into something where direct substitution actually works.
Algebra Reminder: Every "impossible" limit problem becomes possible once you find the right algebraic technique to simplify it.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Piecewise functions are where limits get really interesting because different rules apply on different sides of a point. To find the limit at a boundary point, you need to check what happens approaching from both the left and right sides.
Calculate the left-hand limit using the piece that applies when x is less than your target value. Then find the right-hand limit using the piece that applies when x is greater than your target. If these two values match, that's your limit!
Here's the crucial point: even if the piecewise function has a specific value defined at the boundary point, that doesn't affect the limit. The limit only cares about the approaching behavior, not the actual value at that spot.
Piecewise Strategy: When in doubt, always check both sides separately - piecewise functions love to have different behaviors on each side!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Pre-Calculus