4. Finding one-sided limits (both left-hand and right-hand) first, evaluateeach limit below (you may respond as – o or o when appropriate):2x+2a) limx-2 (x+2)2b) limx2+x+2X-1x-1

Here given as (a) limx→-22x+2x+22 find left-hand limit L.H.S=limh→02(-2-h)+2-2-h+22…

Answered: 4. Finding one-sided limits (both…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

4. Finding one-sided limits (both left-hand and right-hand) first, evaluate
each limit below (you may respond as – o or o when appropriate):
2x+2
a) lim
x-2 (x+2)2
b) lim
x2+x+2
X-1
x-1

Expert Solution

Step 1

Here given as

(a)

$\lim_{x \to - 2} \frac{2 x + 2}{{(x + 2)}^{2}}$

find left-hand limit

$L . H . S = \lim_{h \to 0} \frac{2 (- 2 - h) + 2}{{(- 2 - h + 2)}^{2}} = \lim_{h \to 0} \frac{- 4 - 2 h + 2}{h^{2}} = \lim_{h \to 0} \frac{- 2 (1 + h)}{h^{2}} = - \infty$

Step 2

Find Right-hand side

$R . H . S = \lim_{h \to 0} \frac{2 (- 2 + h) + 2}{{(- 2 + h + 2)}^{2}} = \lim_{h \to 0} \frac{- 4 + 2 h + 2}{h^{2}} = \lim_{h \to 0} \frac{- 2 (1 + h)}{h^{2}} = - \infty$

(b)

$\lim_{x \to 1} \frac{x^{2} + x + 2}{x - 1}$

Find LHS as

$L . H . S = \lim_{h \to 0} \frac{{(1 - h)}^{2} + (1 - h) + 2}{(1 - h - 1)} = \lim_{h \to 0} \frac{1 + h^{2} - 2 h + 1 - h + 2}{h^{2}} = \lim_{h \to 0} \frac{h^{2} - 3 h + 4}{h^{2}} = \infty$

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