Computing Limits: Methods and Examples 2.3

2.3: Computing Limits

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach specific values. Understanding how to compute limits is essential for analyzing continuity, derivatives, and integrals.

Polynomial Functions

Limits of polynomial functions are straightforward because polynomials are continuous everywhere.

• Key Point: For a polynomial $p(x)$, $\lim_{x \to a} p(x) = p(a)$.

• Example: $\lim_{x \to -6} (x^2 + 9x + 4) = (-6)^2 + 9(-6) + 4 = 36 - 54 + 4 = -14$

Rational Functions

Rational functions are quotients of polynomials. The computation of their limits depends on the behavior of the numerator and denominator at the point of interest.

• Key Point: $\lim_{x \to a} \frac{p(x)}{q(x)}$

• If $q(a) \neq 0$, then $\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}$

• If $q(a) = 0$:

  • If $p(a) = 0$, simplify $\frac{p(x)}{q(x)}$ and then evaluate the limit.

  • If $p(a) \neq 0$, the limit is infinite (see section 2.4).

• Example: Given $f(x) = \frac{x-2}{x^2-2x}$, $p(x) = x-2$, $q(x) = x^2-2x$:

  • $\lim_{x \to 1} \frac{x-2}{x^2-2x} = \frac{1-2}{1^2-2 \cdot 1} = \frac{-1}{-1} = 1$

  • $\lim_{x \to 2} \frac{x-2}{x^2-2x} = \lim_{x \to 2} \frac{x-2}{x(x-2)} = \lim_{x \to 2} \frac{1}{x} = \frac{1}{2}$

  • $\lim_{x \to 0} \frac{x-2}{x^2-2x}$: $q(0)=0$, $p(0)\neq 0$; infinite limit.

Factoring and Simplifying Limits

When direct substitution yields an indeterminate form (such as $0/0$), factor and simplify the expression before evaluating the limit.

• Example: $\lim_{x \to 2} \frac{x^2-6x+8}{x^2-4}$

• Factor numerator and denominator: $\frac{(x-2)(x-4)}{(x-2)(x+2)}$

• Simplify: $\lim_{x \to 2} \frac{x-4}{x+2} = \frac{2-4}{2+2} = \frac{-2}{4} = -\frac{1}{2}$

Limits Involving the Difference Quotient

The difference quotient is fundamental in defining the derivative. Limits involving this form often require algebraic manipulation.

• Example: $\lim_{h \to 0} \frac{(10+h)^2 - 100}{h}$

• Expand numerator: $100 + 20h + h^2 - 100 = 20h + h^2$

• Simplify: $\lim_{h \to 0} \frac{h(20+h)}{h} = \lim_{h \to 0} (20 + h) = 20$

Limits Involving Radicals

When limits involve square roots, rationalizing the numerator or denominator can help resolve indeterminate forms.

• Example: $\lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1}$

• Multiply numerator and denominator by $\sqrt{x} + 1$:

• $\lim_{x \to 1} \frac{(\sqrt{x} - 1)(\sqrt{x} + 1)}{(x - 1)(\sqrt{x} + 1)} = \lim_{x \to 1} \frac{x - 1}{(x - 1)(\sqrt{x} + 1)} = \lim_{x \to 1} \frac{1}{\sqrt{x} + 1} = \frac{1}{2}$

• Example: $\lim_{x \to -42} \frac{x - 42}{\sqrt{x} - \sqrt{42}}$

• Rationalize denominator: $\lim_{x \to -42} \frac{(x - 42)(\sqrt{x} + \sqrt{42})}{(\sqrt{x} - \sqrt{42})(\sqrt{x} + \sqrt{42})}$

• Simplify: $\lim_{x \to -42} (\sqrt{x} + \sqrt{42}) = 2\sqrt{42}$

Limits of Piecewise Functions

Piecewise functions may have different definitions on different intervals. Evaluating limits at points where the definition changes requires considering left- and right-sided limits.

• Example: $f(x) = \begin{cases} -2x + 4 & \text{if } x \leq 1 \\ \sqrt{x-1} & \text{if } x > 1 \end{cases}$

• Left-sided limit at $x=1$: $\lim_{x \to 1^-} f(x) = -2(1) + 4 = 2$

• Right-sided limit at $x=1$: $\lim_{x \to 1^+} f(x) = \sqrt{1-1} = 0$

• Since left and right limits are not equal, $\lim_{x \to 1} f(x)$ does not exist (DNE).

• Example: $f(x) = \begin{cases} 0 & x \leq -4 \\ \sqrt{16 - x^2} & -4 < x < 4 \\ 2x & x \geq 4 \end{cases}$

• Left-sided limit at $x=-4$: $\lim_{x \to -4^-} f(x) = 0$

• Right-sided limit at $x=-4$: $\lim_{x \to -4^+} f(x) = \sqrt{16 - (-4)^2} = 0$

• At $x=4$: $\lim_{x \to 4^-} f(x) = \sqrt{16 - 16} = 0$; $\lim_{x \to 4^+} f(x) = 2 \cdot 4 = 8$

• Since left and right limits at $x=4$ are not equal, $\lim_{x \to 4} f(x)$ DNE.

Limit Laws and Operations

Limits can be combined using algebraic operations, provided the individual limits exist.

• 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: $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$

• Power: $\lim_{x \to a} [g(x)]^n = [\lim_{x \to a} g(x)]^n$

• Example: Suppose $\lim_{x \to 2} f(x) = 4$, $\lim_{x \to 2} g(x) = 5$, $\lim_{x \to 2} h(x) = 8$:

  • $\lim_{x \to 2} \frac{f(x) - g(x)}{h(x)} = \frac{4 - 5}{8} = -\frac{1}{8}$

  • $\lim_{x \to 2} [6f(x)g(x) + h(x)] = 6 \cdot 4 \cdot 5 + 8 = 120 + 8 = 128$

  • $\lim_{x \to 2} [g(x)]^3 = 5^3 = 125$

Summary Table: Types of Limits and Methods

Function Type	Method	Example
Polynomial	Direct substitution	$\lim_{x \to a} p(x) = p(a)$
Rational ($q(a) \neq 0$)	Direct substitution	$\lim_{x \to 1} \frac{x-2}{x^2-2x} = 1$
Rational ($q(a) = 0, p(a) = 0$)	Simplify/factor, then substitute	$\lim_{x \to 2} \frac{x^2-6x+8}{x^2-4} = -\frac{1}{2}$
Rational ($q(a) = 0, p(a) \neq 0$)	Infinite limit	$\lim_{x \to 0} \frac{x-2}{x^2-2x}$
Radical	Rationalize numerator/denominator	$\lim_{x \to 1} \frac{\sqrt{x}-1}{x-1} = \frac{1}{2}$
Piecewise	Evaluate left/right limits	$\lim_{x \to 1^-} f(x)$, $\lim_{x \to 1^+} f(x)$

Additional info: These notes cover the main computational techniques for limits, including direct substitution, factoring, rationalizing, and handling piecewise functions. Mastery of these methods is essential for further topics in calculus, such as continuity and differentiation.