Answered: What are the Limit Laws?

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter4: Polynomial And Rational Functions

Section4.2: Polynomial Functions

Problem 96E: What is the purpose of the Intermediate Value Theorem?

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What are the Limit Laws?

Expert Solution

Step 1

Consider the following limits:

$\lim_{x \to a} f (x) = A, \lim_{x \to a} g (x) = B$

Sum law for limits:

$\begin{array}{rcl} \lim_{x \to a} [f (x) + g (x)] & = & \lim_{x \to a} f (x) + \lim_{x \to a} g (x) \\ = & A + B \end{array}$

Difference law for limits:

$\begin{array}{rcl} \lim_{x \to a} [f (x) - g (x)] & = & \lim_{x \to a} f (x) - \lim_{x \to a} g (x) \\ = & A - B \end{array}$

Step 2

Product law for limits:

$\begin{array}{rcl} \lim_{x \to a} [f (x) \times g (x)] & = & \lim_{x \to a} f (x) \times \lim_{x \to a} g (x) \\ = & A \times B \end{array}$

Quotient law for limits:

$\begin{array}{rcl} \lim_{x \to a} [\frac{f (x)}{g (x)}] & = & \frac{\lim_{x \to a} f (x)}{\lim_{x \to a} g (x)} \\ = & \frac{A}{B} f o r B \neq 0 \end{array}$

Constant multiple law for limits:

$\begin{array}{rcl} \lim_{x \to a} c f (x) & = & c \lim_{x \to a} f (x) \\ = & c A \end{array}$

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