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Mastering Calculus A: Practice Problems on Limits

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Shreya Bedi

11/28/2025

Calculus 1

Calculus A: Limits Practice Problems

Subjects

Calculus 1

Limits and Continuity

Connecting Limits at Infinity and Horizontal Asymptotes

65

•

Nov 28, 2025

•

7 pages

Mastering Calculus A: Practice Problems on Limits

user profile picture

Shreya Bedi

@shreyabedi_oyaa

Calculus limits are all about understanding what happens to functions... Show more

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12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

Evaluating Indeterminate Forms

When you see a limit that gives 00\frac{0}{0} at first, don't panic! This is just an indeterminate form telling you to dig deeper.

For example, in lim⁡x→4x2+x−20x−4\lim_{x \to 4} \frac{x^2+x-20}{x-4}, we get 00\frac{0}{0} initially. The trick is to factor the numerator: (x+5)(x−4)(x+5)(x-4), which lets us cancel the (x−4)(x-4) term. The limit simplifies to x+5x+5 evaluated at x=4x=4, giving us 9.

💡 Quick Tip: When a limit gives you 00\frac{0}{0}, try factoring or using algebraic manipulation to cancel common terms before evaluating.

With difference quotients like lim⁡h→0(−9+h)2−81h\lim_{h \to 0} \frac{(-9+h)^2 - 81}{h}, expand and simplify first: h2−18hh=h−18\frac{h^2-18h}{h} = h-18. When hh approaches 0, this becomes 0−18=−180-18 = -18.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

Understanding Infinite Limits

Functions can behave dramatically when approaching certain values - shooting up to infinity, diving down to negative infinity, or approaching a specific value.

When evaluating limits with fractions where the denominator approaches zero, the limit often goes to infinity. For example, lim⁡x→56−x(x−5)\lim_{x \to 5} \frac{6-x}{(x-5)} approaches negative infinity because as xx gets closer to 5, the denominator gets extremely small while the numerator approaches 1.

🔍 Remember: When the denominator approaches zero and the numerator doesn't, your limit will head toward infinity (positive or negative depending on the signs).

For rational functions with large inputs, divide both numerator and denominator by the highest power of the variable. In lim⁡t→∞7t+t2t−t2\lim_{t \to \infty} \frac{7t + t^2}{t - t^2}, dividing by t2t^2 simplifies it to lim⁡t→∞7/t+11/t−1\lim_{t \to \infty} \frac{7/t + 1}{1/t - 1}, which equals -1 as tt approaches infinity.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

Limits and Continuity

Evaluating complex limits becomes easier when you break them down using limit properties. For example, lim⁡x→2(5x2−3x−3)\lim_{x \to 2} (5x^2 - 3x - 3) can be calculated by evaluating each term separately and then combining: 5(2)2−3(2)−3=20−6−3=115(2)^2 - 3(2) - 3 = 20 - 6 - 3 = 11.

Vertical asymptotes (VA) occur when a function's denominator equals zero, making the function undefined. Horizontal asymptotes (HA) appear when x approaches infinity and the function approaches a constant value.

📊 Visual Aid: Think of asymptotes as "barriers" the function gets extremely close to but never touches or crosses.

Average velocity is calculated as S(t2)−S(t1)t2−t1\frac{S(t_2) - S(t_1)}{t_2 - t_1}, which is the change in position divided by the change in time. For instance, if a position changes from 78.7 to 20.1 in 2 seconds, the average velocity is 78.7−20.12=29.3\frac{78.7 - 20.1}{2} = 29.3 ft/s.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

Special Limit Cases

When evaluating limits involving square roots, rationalizing can help. For expressions like lim⁡h→064+h−8h\lim_{h \to 0} \frac{\sqrt{64 + h} - 8}{h}, multiplying both numerator and denominator by the conjugate helps eliminate the indeterminate form.

For functions like f(x)=3x2−x4f(x) = 3x^2 - x^4, understanding end behavior is crucial. As xx approaches infinity, the −x4-x^4 term dominates, making the limit approach negative infinity. Writing it as x2(3−x2)x^2(3 - x^2) helps visualize this behavior.

🧠 Think About It: The highest-power term usually determines how a function behaves as x approaches infinity.

When analyzing limits of piecewise functions, you need to evaluate each piece separately. For a function defined differently for x<0x < 0, 0≤x≤10 \leq x \leq 1, and x>1x > 1, examine the limit from both directions when approaching boundary points like x=0x = 0 or x=1x = 1.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

Limit Applications and Properties

When dealing with combined functions, apply limit properties. For instance, lim⁡x→0x+3x24x−1/x\lim_{x \to 0} \frac{\sqrt{x} + 3x^2}{4x - 1/x} can be simplified by analyzing which terms dominate as x approaches zero.

The squeeze theorem is incredibly useful when functions are bounded. If 3x−1≤f(x)≤x2−3x+83x - 1 \leq f(x) \leq x^2 - 3x + 8 and both bounds equal 8 when x=3x = 3, then lim⁡x→3f(x)=8\lim_{x \to 3} f(x) = 8.

🔑 Key Insight: Limit properties let you break complex expressions into manageable pieces. Remember that lim⁡x→a[f(x)+g(x)]=lim⁡x→af(x)+lim⁡x→ag(x)\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x).

When analyzing function behavior near vertical asymptotes, check the limit from both directions. For lim⁡x→5x2−5xx2−10x+25\lim_{x \to 5} \frac{x^2 - 5x}{x^2 - 10x + 25}, simplifying to xx−5\frac{x}{x-5} shows that the limit approaches +∞+\infty from the right and −∞-\infty from the left.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

Working with Rational Functions

When evaluating limits of rational functions as x approaches infinity, focus on the terms with the highest powers in both numerator and denominator. This determines the end behavior of the function.

For complex fractions like lim⁡x→∞(5x2+1)(x−7)2(x2+x)\lim_{x \to \infty} \frac{(5x^2+1)}{(x-7)^2(x^2+x)}, divide every term by the highest power of x in this case $x^4$ to simplify the expression.

⚡ Power Tip: When x approaches infinity, terms with lower powers become negligible compared to those with higher powers.

This approach transforms complicated rational functions into manageable expressions where most terms approach zero, leaving only the ratio of the coefficients of the highest-power terms.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X


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Subjects

Calculus 1

Limits and Continuity

Connecting Limits at Infinity and Horizontal Asymptotes

 

Calculus 1

•

65

•

Nov 28, 2025

•

7 pages

Mastering Calculus A: Practice Problems on Limits

user profile picture

Shreya Bedi

@shreyabedi_oyaa

Calculus limits are all about understanding what happens to functions when they approach certain values. Think of limits as a mathematical zoom lens, letting you see exactly where a function is heading, even at tricky points where the function might... Show more

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

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Evaluating Indeterminate Forms

When you see a limit that gives 00\frac{0}{0} at first, don't panic! This is just an indeterminate form telling you to dig deeper.

For example, in lim⁡x→4x2+x−20x−4\lim_{x \to 4} \frac{x^2+x-20}{x-4}, we get 00\frac{0}{0} initially. The trick is to factor the numerator: (x+5)(x−4)(x+5)(x-4), which lets us cancel the (x−4)(x-4) term. The limit simplifies to x+5x+5 evaluated at x=4x=4, giving us 9.

💡 Quick Tip: When a limit gives you 00\frac{0}{0}, try factoring or using algebraic manipulation to cancel common terms before evaluating.

With difference quotients like lim⁡h→0(−9+h)2−81h\lim_{h \to 0} \frac{(-9+h)^2 - 81}{h}, expand and simplify first: h2−18hh=h−18\frac{h^2-18h}{h} = h-18. When hh approaches 0, this becomes 0−18=−180-18 = -18.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

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Understanding Infinite Limits

Functions can behave dramatically when approaching certain values - shooting up to infinity, diving down to negative infinity, or approaching a specific value.

When evaluating limits with fractions where the denominator approaches zero, the limit often goes to infinity. For example, lim⁡x→56−x(x−5)\lim_{x \to 5} \frac{6-x}{(x-5)} approaches negative infinity because as xx gets closer to 5, the denominator gets extremely small while the numerator approaches 1.

🔍 Remember: When the denominator approaches zero and the numerator doesn't, your limit will head toward infinity (positive or negative depending on the signs).

For rational functions with large inputs, divide both numerator and denominator by the highest power of the variable. In lim⁡t→∞7t+t2t−t2\lim_{t \to \infty} \frac{7t + t^2}{t - t^2}, dividing by t2t^2 simplifies it to lim⁡t→∞7/t+11/t−1\lim_{t \to \infty} \frac{7/t + 1}{1/t - 1}, which equals -1 as tt approaches infinity.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

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Limits and Continuity

Evaluating complex limits becomes easier when you break them down using limit properties. For example, lim⁡x→2(5x2−3x−3)\lim_{x \to 2} (5x^2 - 3x - 3) can be calculated by evaluating each term separately and then combining: 5(2)2−3(2)−3=20−6−3=115(2)^2 - 3(2) - 3 = 20 - 6 - 3 = 11.

Vertical asymptotes (VA) occur when a function's denominator equals zero, making the function undefined. Horizontal asymptotes (HA) appear when x approaches infinity and the function approaches a constant value.

📊 Visual Aid: Think of asymptotes as "barriers" the function gets extremely close to but never touches or crosses.

Average velocity is calculated as S(t2)−S(t1)t2−t1\frac{S(t_2) - S(t_1)}{t_2 - t_1}, which is the change in position divided by the change in time. For instance, if a position changes from 78.7 to 20.1 in 2 seconds, the average velocity is 78.7−20.12=29.3\frac{78.7 - 20.1}{2} = 29.3 ft/s.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

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Special Limit Cases

When evaluating limits involving square roots, rationalizing can help. For expressions like lim⁡h→064+h−8h\lim_{h \to 0} \frac{\sqrt{64 + h} - 8}{h}, multiplying both numerator and denominator by the conjugate helps eliminate the indeterminate form.

For functions like f(x)=3x2−x4f(x) = 3x^2 - x^4, understanding end behavior is crucial. As xx approaches infinity, the −x4-x^4 term dominates, making the limit approach negative infinity. Writing it as x2(3−x2)x^2(3 - x^2) helps visualize this behavior.

🧠 Think About It: The highest-power term usually determines how a function behaves as x approaches infinity.

When analyzing limits of piecewise functions, you need to evaluate each piece separately. For a function defined differently for x<0x < 0, 0≤x≤10 \leq x \leq 1, and x>1x > 1, examine the limit from both directions when approaching boundary points like x=0x = 0 or x=1x = 1.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

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Limit Applications and Properties

When dealing with combined functions, apply limit properties. For instance, lim⁡x→0x+3x24x−1/x\lim_{x \to 0} \frac{\sqrt{x} + 3x^2}{4x - 1/x} can be simplified by analyzing which terms dominate as x approaches zero.

The squeeze theorem is incredibly useful when functions are bounded. If 3x−1≤f(x)≤x2−3x+83x - 1 \leq f(x) \leq x^2 - 3x + 8 and both bounds equal 8 when x=3x = 3, then lim⁡x→3f(x)=8\lim_{x \to 3} f(x) = 8.

🔑 Key Insight: Limit properties let you break complex expressions into manageable pieces. Remember that lim⁡x→a[f(x)+g(x)]=lim⁡x→af(x)+lim⁡x→ag(x)\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x).

When analyzing function behavior near vertical asymptotes, check the limit from both directions. For lim⁡x→5x2−5xx2−10x+25\lim_{x \to 5} \frac{x^2 - 5x}{x^2 - 10x + 25}, simplifying to xx−5\frac{x}{x-5} shows that the limit approaches +∞+\infty from the right and −∞-\infty from the left.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

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Working with Rational Functions

When evaluating limits of rational functions as x approaches infinity, focus on the terms with the highest powers in both numerator and denominator. This determines the end behavior of the function.

For complex fractions like lim⁡x→∞(5x2+1)(x−7)2(x2+x)\lim_{x \to \infty} \frac{(5x^2+1)}{(x-7)^2(x^2+x)}, divide every term by the highest power of x in this case $x^4$ to simplify the expression.

⚡ Power Tip: When x approaches infinity, terms with lower powers become negligible compared to those with higher powers.

This approach transforms complicated rational functions into manageable expressions where most terms approach zero, leaving only the ratio of the coefficients of the highest-power terms.

12. () = x² + x -20 |x-1| g(x) x-7* lim g(x) = 16+ 4-20 14-4) g(x) = (x + 5)(x-7) (+47 g(x) = (x+5) lim gex] [9] X-4' lim y(x) = (x+5) (x) X

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We thought you’d never ask...

What is the Knowunity AI companion?

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.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

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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

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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

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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