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: [latex]\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)[/latex]
    • Product: [latex]\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)[/latex]
    • Quotient: [latex]\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}[/latex], if [latex]\lim_{x \to a} g(x) \neq 0[/latex]
    • Power: [latex]\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n[/latex]
  • Polynomial and Rational Function Limits:
    • [latex]\lim_{x \to a} p(x) = p(a)[/latex] for polynomials
    • [latex]\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}[/latex] for rational functions, if [latex]q(a) \neq 0[/latex]
  • 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:
    • [latex]\frac{0}{0}[/latex], [latex]\frac{\infty}{\infty}[/latex], [latex]0 \cdot \infty[/latex], etc.
    • Require special techniques for evaluation

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

Evaluate [latex]\underset{x\to -2}{\lim}(3x^3-2x+7)[/latex].

Evaluate [latex]\underset{x\to -3}{\lim}\dfrac{x^2+4x+3}{x^2-9}[/latex]

Evaluate [latex]\underset{x\to 5}{\lim}\dfrac{\sqrt{x-1}-2}{x-5}[/latex]

Evaluate [latex]\underset{x\to -3}{\lim}\dfrac{\frac{1}{x+2}+1}{x+3}[/latex]

Evaluate [latex]\underset{x\to 3}{\lim}\left(\dfrac{1}{x-3}-\dfrac{4}{x^2-2x-3}\right)[/latex]

The Squeeze Theorem

The Main Idea 

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

Evaluate the following limit using the Squeeze Theorem:

[latex]\lim_{\theta \to 0} \frac{\theta - \sin \theta}{\theta^3}[/latex]

Evaluate the following limit:

[latex]\lim_{x \to 0} x \cos(\frac{1}{x})[/latex]