Determining limits analytically Determine the following limits.a. limxS1 x - 1|b. limxS1 x - 1|c. limxSi x - 1

a. limx→1+xx-1=limx→1+xx-1 ∵x-1=x-1 if x≥1-x-1 if x<1=11-1=10=∞ b.…

Answered: Determining limits analytically…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.2: Exponential Functions

Problem 20E

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Question

Determining limits analytically Determine the following limits.
a. lim
xS1 x - 1|
b. lim
xS1 x - 1|
c. lim
xSi x - 1

Expert Solution

Step 1

$\begin{array}{rcl} \lim_{x \to 1^{+}} \frac{x}{|x - 1|} & = & \lim_{x \to 1^{+}} (\frac{x}{x - 1}) [∵ |x - 1| = \{\begin{cases} x - 1 if x \geq 1 \\ - (x - 1) if x < 1 \end{cases}] \\ = & \frac{1}{1 - 1} \\ = & \frac{1}{0} \\ = & \infty \end{array}$

$\begin{array}{rcl} \lim_{x \to 1^{-}} \frac{x}{|x - 1|} & = & \lim_{x \to 1^{-}} (\frac{x}{- (x - 1)}) [∵ |x - 1| = \{\begin{cases} x - 1 if x \geq 1 \\ - (x - 1) if x < 1 \end{cases}] \\ = & \lim_{x \to 1^{-}} (\frac{x}{- x + 1}) \\ = & \frac{1}{- 1 + 1} \\ = & \frac{1}{0} \\ = & \infty \end{array}$