QuestionFind the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . If it doesn't exist, enter DNE.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit of the function as x approaches infinity. . The function is $\sqrt{25 x^{}+x}-5 x$

STEP 2

The function is in the form of $\sqrt{a x^{2}+b x}-c x$ which is indeterminate of the form0/0 or ∞/∞ as x approaches infinity. To solve such limits, we can use the method of rationalization.

STEP 3

Multiply and divide the function by its conjugate. The conjugate of $\sqrt{25 x^{2}+x}-5 x$ is $\sqrt{25 x^{2}+x}+5 x$ .
$\lim{x \rightarrow \infty}\frac{\sqrt{25 x^{2}+x}-5 x}{1} \times \frac{\sqrt{25 x^{2}+x}+5 x}{\sqrt{25 x^{2}+x}+5 x}$

STEP 4

implify the numerator using the formula (a-b)(a+b) = a^2 - b^2.
$\lim{x \rightarrow \infty}\frac{(25 x^{2}+x) - ( x)^2}{\sqrt{25 x^{2}+x}+ x}$

STEP 5

implify the numerator.
$\lim{x \rightarrow \infty}\frac{25 x^{2}+x -25 x^2}{\sqrt{25 x^{2}+x}+5 x}$

STEP 6

implify further.
$\lim{x \rightarrow \infty}\frac{x}{\sqrt{25 x^{2}+x}+5 x}$

STEP 7

Divide the numerator and denominator by x.
$\lim{x \rightarrow \infty}\frac{1}{\sqrt{25 +1/x}+5}$

STEP 8

As x approaches infinity,1/x approaches0.
$\lim{x \rightarrow \infty}\frac{1}{\sqrt{25 +0}+5}$

STEP 9

implify the denominator.
$\lim{x \rightarrow \infty}\frac{}{\sqrt{25}+5}$

STEP 10

Calculate the limit.
$\lim{x \rightarrow \infty}\frac{}{5+5} = \frac{}{10}$ So, the limit of the function as x approaches infinity is/10.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms