Find the limit algebraically.lim (V25x2 + x – 5x)-

the given limit is: limx→∞25x2+x-5x we have to find the value of the limit.

Answered: Find the limit algebraically. lim…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.4: Applications Of Eigenvalues And Eigenvectors

Problem 10E: Find the limit if it exists of Anx1 as n approaches infinity, where A=[02120], and x1=[aa]

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Question

Find the limit algebraically.
lim (V25x2 + x – 5x)
-

Expert Solution

Step 1

the given limit is:

$\lim_{x \to \infty} (\sqrt{25 x^{2} + x} - 5 x)$

we have to find the value of the limit.

Step 2

the given limit is:

$\lim_{x \to \infty} (\sqrt{25 x^{2} + x} - 5 x)$

rationalize the expression.

therefore,

$\begin{array}{rcl} \lim_{x \to \infty} (\sqrt{25 x^{2} + x} - 5 x) & = & \lim_{x \to \infty} (\sqrt{25 x^{2} + x} - 5 x) \times \frac{\sqrt{25 x^{2} + x} + 5 x}{\sqrt{25 x^{2} + x} + 5 x} \\ = & \lim_{x \to \infty} \frac{{(\sqrt{25 x^{2} + x})}^{2} - {(5 x)}^{2}}{\sqrt{25 x^{2} + x} + 5 x} (b e c a u s e (a - b) (a + b) = a^{2} - b^{2}) \\ = & \lim_{x \to \infty} \frac{(25 x^{2} + x) - 25 x^{2}}{\sqrt{25 x^{2} + x} + 5 x} \\ = & \lim_{x \to \infty} \frac{25 x^{2} - 25 x^{2} + x}{\sqrt{25 x^{2} + x} + 5 x} \\ = & \lim_{x \to \infty} \frac{x}{\sqrt{25 x^{2} + x} + 5 x} \\ = & \lim_{x \to \infty} \frac{x}{\sqrt{x^{2} (25 + \frac{x}{x^{2}})} + 5 x} \\ = & \lim_{x \to \infty} \frac{x}{x \sqrt{25 + \frac{1}{x}} + 5 x} \\ = & \lim_{x \to \infty} \frac{x}{x (\sqrt{25 + \frac{1}{x}} + 5)} \\ = & \lim_{x \to \infty} \frac{1}{\sqrt{25 + \frac{1}{x}} + 5} \\ = & \frac{1}{\sqrt{25 + \frac{1}{\infty}} + 5} \\ = & \frac{1}{\sqrt{25 + 0} + 5} \\ = & \frac{1}{5 + 5} \\ = & \frac{1}{10} \end{array}$

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