Understanding Limits

A limit represents the value a function approaches as the input gets infinitely close to a specific number. Written as $\lim_{x \to a} f(x)$, it shows what happens to f(x) as x gets closer and closer to a—without necessarily being exactly a.

Sometimes a function's limit at a point exists even when the function itself isn't defined there. Consider $y = \frac{(x-1)^2}{(x-1)}$. At x = 1, we get $\frac{0}{0}$ which is undefined. But what happens as we approach 1 from both sides?

By creating a table of values with x-values getting increasingly close to 1 (like 0.999 or 1.001), we can see the y-values converge to 0. This tells us that $\lim_{x\to 1} \frac{(x-1)^2}{(x-1)} = 0$, even though f(1) doesn't exist.

💡 Remember: The limit of a function at a point can exist even when the function value doesn't. Always check by approaching from both sides!

Indeterminate Forms and Simplification

When evaluating limits, you might encounter expressions like $\frac{0}{0}$—this is an indeterminate form. While these expressions don't have well-defined values, we can often simplify them to find the limit.

For example, in our previous function $\frac{(x-1)^2}{(x-1)}$, we can simplify to $x-1$ for all values except x = 1. This simplification helps us determine that the limit equals 0.

It's important to distinguish between indeterminate forms and undefined expressions. When an expression gives $\frac{1}{0}$, it's not indeterminate—it's undefined, and the limit doesn't exist.

Let's try an example: For $\lim_{x\to 1} \frac{x^2-1}{x-1}$, we can factor the numerator to get $\frac{(x-1)(x+1)}{x-1}$. Simplifying gives us $(x+1)$. Therefore, $\lim_{x\to 1} \frac{x^2-1}{x-1} = 2$.

Infinite Limits

Some limits grow without bound as x approaches a specific value. These are called infinite limits.

Consider $\lim_{x\to 0} \frac{1}{x^2}$. As x gets closer to 0 from either side, the function values grow incredibly large. Looking at a table of values:

We can see the function approaches positive infinity from both sides. We write this as $\lim_{x \to 0} \frac{1}{x^2} = +\infty$, though technically this means the limit doesn't exist as infinity is not a real number.

One-sided limits are particularly important with functions like $\frac{1}{x}$. As x approaches 0 from the positive side $x \to 0^+$, the limit is $+\infty$. From the negative side $x \to 0^-$, it's $-\infty$.

🔑 Key insight: When the left and right limits disagree like with $\frac{1}{x}$ at x = 0, the limit doesn't exist!

One-Sided Limits

When evaluating tricky limits, examining what happens as we approach from each side can be crucial. This is especially important for functions like $\frac{1}{x}$.

Creating a value table for $\lim_{x \to 0} \frac{1}{x}$:

We observe that as x approaches 0 from the right, the function approaches $+\infty$, but as x approaches from the left, it approaches $-\infty$. Since these one-sided limits disagree, $\lim_{x \to 0} \frac{1}{x}$ does not exist.

This highlights an important principle: for a limit to exist, both the left-side limit and right-side limit must exist and be equal. If they differ, even if both approach infinity, the limit doesn't exist.

Continuity of Functions

A function is continuous at a point when two conditions are met: (1) the limit exists at that point, and (2) the function value at that point equals the limit. Mathematically, a function is continuous at x = a if $\lim_{x\to a} f(x) = f(a)$ and f(a) is defined.

Continuity means a function's graph has no breaks, jumps, or holes. For example, $y = \frac{1}{x}$ is discontinuous at x = 0 because f(0) is undefined (you can't divide by zero).

Functions like $y = x^2$ are continuous everywhere because the limit exists and equals the function value at every point. Meanwhile, $y = \sqrt{1-x}$ is continuous only where x < 1, as the function isn't defined for x ≥ 1.

📝 Quick tip: To check continuity, ask yourself: "Can I draw this function without lifting my pencil from the paper?" If not, there's a discontinuity!

Types of Discontinuities

Discontinuities come in different forms. An infinite discontinuity occurs when the function approaches infinity at a point, like with $y = \frac{1}{x}$ at x = 0.

When analyzing continuity, always check both conditions:

1. Does the limit exist at the point?
2. Is the function defined at the point?
3. Does the limit equal the function value?

For example, in $y = \frac{1}{x}$ at x = 0, the limit doesn't exist and the function isn't defined—double violation!

Limits at Infinity

Limits at infinity examine function behavior as x grows arbitrarily large (positive or negative). These are written as $\lim_{x \to +\infty} f(x)$ or $\lim_{x \to -\infty} f(x)$.

When evaluating these limits, pay attention to the term with the highest exponent, as it dominates the function's behavior as x grows large. For rational functions, compare the degrees of the numerator and denominator polynomials.

Evaluating Limits at Infinity

For simple rational functions like $\lim_{x \to \infty} \frac{1}{x}$, creating a table helps show the trend:

As x grows larger, the function values approach 0. Thus, $\lim_{x \to \infty} \frac{1}{x} = 0$.

This illustrates an important principle: when a variable appears in the denominator and approaches infinity, the fraction approaches zero. You can think of it as dividing by an increasingly large number, making the result increasingly tiny.

For more complex expressions, you'll need to identify which terms matter most as x grows. The term with the highest power of x will ultimately determine the function's behavior.

🧠 Mental shortcut: For rational functions as x approaches infinity, only the highest-power terms in the numerator and denominator matter!

Polynomial Behavior at Infinity

For polynomial functions like $\lim_{x\to \infty} (x^3 +3x^2-6x)$, the term with the highest exponent (x³) dominates as x gets very large. Since this term has a positive coefficient, the limit is $+\infty$.

The sign of the limit depends on two factors:

1. The coefficient of the highest-degree term (positive or negative)
2. Whether the highest exponent is even or odd

This handy table summarizes the behavior:

For example, with $x^3$ (odd exponent, positive coefficient), the limits are $\lim_{x\to\infty} x^3 = +\infty$ and $\lim_{x\to-\infty} x^3 = -\infty$.

Remembering these patterns can save valuable time during tests!

Rational Functions at Infinity

When dealing with rational functions at infinity, compare the degrees of the numerator and denominator. For $\lim_{x \to \infty} \frac{5x^3+2x-4}{3x^3+3}$, both have the same highest power (x³).

In such cases, the limit equals the ratio of the coefficients of the highest power terms:

$\lim_{x \to \infty} \frac{5x^3+2x-4}{3x^3+3} = \lim_{x \to \infty} \frac{5x^3}{3x^3} = \frac{5}{3}$

A formal approach is to factor out the highest power:

$\lim_{x \to \infty} \frac{5x^3+2x-4}{3x^3+3} = \lim_{x \to \infty} \frac{x^3(5+\frac{2}{x^2}-\frac{4}{x^3})}{x^3(3+\frac{3}{x^3})} = \lim_{x \to \infty} \frac{5+\frac{2}{x^2}-\frac{4}{x^3}}{3+\frac{3}{x^3}} = \frac{5}{3}$

As x→∞, the terms with x in the denominator approach zero, leaving only the coefficients of the highest powers.

This technique works for all rational functions and saves time during exams.

Rules for Limits at Infinity

The behavior of rational functions at infinity follows these patterns:

1. Numerator degree > denominator degree: The limit is ±∞, determined by the leading terms' signs and whether the power difference is odd or even.

2. Numerator degree < denominator degree: The limit is always 0, regardless of coefficients.

3. Equal degrees: The limit equals the ratio of the leading coefficients $\frac{\text{leading coef. of numerator}}{\text{leading coef. of denominator}}$.

For example:

• $\lim_{x \to \infty} \frac{2x^3}{x} = \infty$ (numerator degree higher)
• $\lim_{x \to \infty} \frac{5x}{3x^2} = 0$ (denominator degree higher)
• $\lim_{x \to \infty} \frac{4x^2}{2x^2} = 2$ (equal degrees)

🌟 Test success tip: Always identify the degrees first! This simple step will immediately tell you which rule to apply and save precious time during your exam.