Existence of limits, one sided limits, and infinity limits.

One sided limits

When $x$ approaches a point $a$ from one side only:

• Left-sided limit: $\lim_{x \to a^-} f(x) = L$
  • means $x$ approaches $a$ with $x < a, f(x) \to L$
• Right-sided limit: $\lim_{x \to a^+} f(x) = L$
  • means $x$ approaches $a$ with $x > a, f(x) \to L$

Examples:

$$\lim_{x \to 2^-} x^2 = 4$$

$$\lim_{x \to 2^+} x^2 = 4$$

$$\lim_{x \to 0^+} \frac{1}{x} = \infty$$

$$\lim_{x \to 0^-} \frac{1}{x} = -\infty$$

$$\lim_{x \to 0^+} \frac{|x|}{x} = \lim_{x \to 0^+} \frac{x}{x} = 1$$

$$\lim_{x \to 0^-} \frac{|x|}{x} = \lim_{x \to 0^-} \frac{-x}{x} = -1$$

Existence of limits

A limit exists if and only if the left-sided limit and the right-sided limit are equal.

$$\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$

Types of non-existence

• Jump discontinuity: $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$.
• Infinite discontinuity: $\lim_{x \to a^+} f(x) = \infty$ and $\lim_{x \to a^-} f(x) = -\infty$ for example.

Examples:

$$\lim_{x \to 0} \frac{|x|}{x} \text{ does not exist because } \lim_{x \to 0^+} \frac{|x|}{x} = 1 \neq \lim_{x \to 0^-} \frac{|x|}{x} = -1$$

$$\lim_{x \to 0} \frac{1}{x} \text{ does not exist because } \lim_{x \to 0^+} \frac{1}{x} = \infty \neq \lim_{x \to 0^-} \frac{1}{x} = -\infty$$

Example

$$f(x) = \begin{cases} x^2 + 1 & \text{if } x \leq 0 \\ x + 1 & \text{if } x > 0 \end{cases}$$

$$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x^2 + 1) = 1$$

$$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x + 1) = 1$$

$$\lim_{x \to 0} f(x) = 1$$

Infinity limits

Vertical Asymptotes

If $f(x)$ grows without bound as $x$ approaches $a$, we write

$$\lim_{x \to a} f(x) = \infty$$

or

$$\lim_{x \to a} f(x) = -\infty$$

This signals a vertical asymptote at $x = a$.

Example:

$$h(x) = \frac{1}{(x-3)^2}$$

As $x \to 3$, denominator $\to 0$, numerator positive $\to$

$$\lim_{x \to 3} h(x) = \infty$$

One-sided blow-up

$$p(x) = \frac{1}{x-1}$$

One‐sided blow‐up:

$$\lim_{x \to 1^-} p(x) = -\infty$$

$$\lim_{x \to 1^+} p(x) = \infty$$

Horizontal Asymptotes

Describing end‐behavior:

$$\lim_{x \to \infty} f(x) = L$$

$$\lim_{x \to \infty} f(x) = L \implies y = L \text{ is a horizontal asymptote on the right}$$

Similarly for $\lim_{x \to -\infty} f(x) = L$.

Examples:

$$\lim_{x \to \infty} \frac{2x^2 + 3}{5x^2 - 1} = \frac{2}{5}$$

$$\lim_{x \to -\infty} \frac{2x^2 + 3}{5x^2 - 1} = \frac{2}{5}$$

Example 2:

$$\lim_{x \to \infty} 2^{-x} = 0$$

Example 3:

$$\lim_{x \to -\infty} 2^{-x} = \infty$$

Exercises

0. Feedback

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1. Check your understanding

2. Evaluate the limit using graphs

Use graphs (Desmos or GeoGebra) to estimate the following limits.

3. Evaluate the limit using algebra

Do the corrective assignment problems in the following link:
solving limits algebraically

(For more problems and videos, see this lesson page and this other lesson)