The Limit Laws: Fresh Take

Recognize and apply basic limit laws to evaluate the limits of functions, including polynomials and rational functions
Evaluate the limit of a function by factoring or by using conjugates
Evaluate the limit of a function by using the squeeze theorem

Evaluating Limits

The Main Idea

Limit Laws:
- Sum/Difference: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$
- Product: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
- Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ , if $\lim_{x \to a} g(x) \neq 0$
- Power: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
Polynomial and Rational Function Limits:
- $\lim_{x \to a} p(x) = p(a)$ for polynomials
- $\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}$ for rational functions, if $q(a) \neq 0$
Techniques for Indeterminate Forms:
- Factoring and Simplifying:
  - Factor polynomials and cancel common terms
- Rationalizing (Multiplying by Conjugate):
  - Used for limits with square roots
- Simplifying Complex Fractions:
  - Combine fractions using LCD
Indeterminate Forms:
- $\frac{0}{0}$ , $\frac{\infty}{\infty}$ , $0 \cdot \infty$ , etc.
- Require special techniques for evaluation

Use the limit laws to evaluate $\underset{x\to 6}{\lim}(2x-1)\sqrt{x+4}$ . In each step, indicate the limit law applied.

$11\sqrt{10}$

Evaluate $\underset{x\to -2}{\lim}(3x^3-2x+7)$ .

$−13$

Evaluate $\underset{x\to -3}{\lim}\dfrac{x^2+4x+3}{x^2-9}$

$\dfrac{1}{3}$

Evaluate $\underset{x\to 5}{\lim}\dfrac{\sqrt{x-1}-2}{x-5}$

$\dfrac{1}{4}$

Evaluate $\underset{x\to -3}{\lim}\dfrac{\frac{1}{x+2}+1}{x+3}$

$−1$

Evaluate $\underset{x\to 3}{\lim}\left(\dfrac{1}{x-3}-\dfrac{4}{x^2-2x-3}\right)$

Don’t forget to factor $x^2-2x-3$ before getting a common denominator.

$\frac{1}{4}$

The Main Idea

The Squeeze Theorem:
- If $g(x) \leq f(x) \leq h(x)$ near $a$ , and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} f(x) = L$
Key Trigonometric Limits:
- $\lim_{\theta \to 0} \sin \theta = 0$
- $\lim_{\theta \to 0} \cos \theta = 1$
- $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$
- $\lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} = 0$
Geometric Intuition:
- For small $\theta$ : $0 < \sin \theta < \theta < \tan \theta$ (in radians)
- It is helpful to visualize the unit circle for sine, cosine, and tangent

Evaluate the following limit using the Squeeze Theorem:

$\lim_{\theta \to 0} \frac{\theta - \sin \theta}{\theta^3}$

Recall that for small $\theta$ : $0 < \sin \theta < \theta$ This means: $0 < \theta - \sin \theta < \theta - 0 = \theta$

Divide all parts by $\theta^3$ (which is positive for small, positive $\theta$ ):

$0 < \frac{\theta - \sin \theta}{\theta^3} < \frac{\theta}{\theta^3} = \frac{1}{\theta^2}$

Now we have bounds for our function:

$0 \leq \frac{\theta - \sin \theta}{\theta^3} \leq \frac{1}{\theta^2}$

As $\theta \to 0$ , both bounds approach $0$ :

$\lim_{\theta \to 0} 0 = 0$ and $\lim_{\theta \to 0} \frac{1}{\theta^2} = \infty$

By the Squeeze Theorem:

$\lim_{\theta \to 0} \frac{\theta - \sin \theta}{\theta^3} = 0$

Evaluate the following limit:

$\lim_{x \to 0} x \cos(\frac{1}{x})$

Note that $\cos(\frac{1}{x})$ oscillates between $-1$ and $1$ as $x$ approaches $0$ .

Therefore, we can bound our function:

$-|x| \leq x \cos(\frac{1}{x}) \leq |x|$

As $x$ approaches $0$ , both $-|x|$ and $|x|$ approach $0$ :

$\lim_{x \to 0} -|x| = \lim_{x \to 0} |x| = 0$

By the Squeeze Theorem:

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