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  • Evaluate following limit: <img alt="\lim_{x\rightarrow \infty }\left ( \sqrt{4x^{2}+7x} -2x\right )" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%20%5Cinfty%20

Evaluate following limit: \lim_{x\rightarrow \infty }\left ( \sqrt{4x^{2}+7x} -2x\right )

Option: 1

\frac{5}{4}


Option: 2

\frac{7}{4}


Option: 3

\frac{1}{4}


Option: 4

\frac{3}{4}


Answers (1)

best_answer

For solving the limits of the type ∞ - ∞, we simply rationalize the given limit, that is multiplying and dividing the limit by its additive inverse.

Additive inverse of \lim_{x\rightarrow \infty }\left ( \sqrt{4x^{2}+7x} -2x\right ) is \lim_{x\rightarrow \infty }\left ( \sqrt{4x^{2}+7x} +2x\right )

Hence,

\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{\left(\sqrt{4 x^2+7 x}-2 x\right)\left(\sqrt{4 x^2+7 x}+2 x\right)}{\left(\sqrt{4 x^2+7 x}+2 x\right)} \\ = & \lim _{x \rightarrow \infty} \frac{4 x^2+7 x-4 x^2}{\left(\sqrt{4 x^2+7 x}+2 x\right)} \end{aligned}

\text { Dividing numerator and denominator by } \mathrm{x} \text { : }

\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{7}{\left(\sqrt{4+\frac{7}{x}}+2\right)} \\ & =\frac{7}{4} \end{aligned}

 

Posted by

himanshu.meshram

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