19-70. Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed realnumbers.34. limI→356. limx² 2x42. Tim (2²-²1)w→1Wx − 31x→1 √√√4x + 5-362. lim64. limCOS I 1x→0 cos²x - 13I|w3|w+3 w² - 7w+12

34. x2 - 2x - 3 = ( x - 3) ( x + 1) limx→3x2-2x-3x-3=limx→3(x-3)(x+1)x-3=limx→3 x + 1 = 3 + 1 = 4…

Answered: 19-70. Evaluating limits Find the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

19-70. Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real
numbers.
34. lim
I→3
56. lim
x² 2x
42. Tim (2²-²1)
w→1
W
x − 3
1
x→1 √√√4x + 5-3
62. lim
64. lim
COS I 1
x→0 cos²x - 1
3
I
|w3|
w+3 w² - 7w+12

Expert Solution

Step 1

34. x² - 2x - 3 = ( x - 3) ( x + 1)

$\lim_{x \to 3} \frac{x^{2} - 2 x - 3}{x - 3} = \lim_{x \to 3} \frac{(x - 3) (x + 1)}{x - 3} = \lim_{x \to 3} x + 1 = 3 + 1 = 4$

42.

$\lim_{w \to 1} (\frac{1}{w^{2} - w} - \frac{1}{w - 1}) = \underset{w \to 1}{\lim (} \frac{1}{w (w - 1)} - \frac{1}{w - 1}) = \lim_{w \to 1} \frac{1 - w}{w (w - 1)} = \lim_{w \to 1} \frac{- 1}{w} = - 1$

56.

$\lim_{x \to 1} \frac{x - 1}{\sqrt{4 x + 5} - 3} = \lim_{x \to 1} \frac{x - 1}{\sqrt{4 x + 5} - 3} \times \frac{\sqrt{4 x + 5} + 3}{\sqrt{4 x + 5} + 3} = \lim_{x \to 1} \frac{(x - 1) (\sqrt{4 x + 5} + 3)}{4 x + 5 - 9} = \lim_{x \to 1} \frac{(x - 1) (\sqrt{4 x + 5} + 3)}{4 (x - 1)} = \lim_{x \to 1} \frac{\sqrt{4 x + 5} + 3}{4} = \frac{6}{4} = \frac{3}{2}$