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  • Find value of given limit: <img alt="\lim _{x \rightarrow \infty}\left(\sqrt{\left(4 x^2+100 x+1\right)}-\sqrt{\left(4 x^2+1\right)}\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?%5C

Find value of given limit: \lim _{x \rightarrow \infty}\left(\sqrt{\left(4 x^2+100 x+1\right)}-\sqrt{\left(4 x^2+1\right)}\right)

Option: 1

25


Option: 2

50


Option: 3

75


Option: 4

100


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{\left(4 x^2+100 x+1\right)}-\sqrt{\left(4 x^2+1\right)}\right) is \lim _{x \rightarrow \infty}\left(\sqrt{\left(4 x^2+100 x+1\right)}+\sqrt{\left(4 x^2+1\right)}\right)

Hence,

\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{\left(\sqrt{\left(4 x^2+100 x+1\right)}-\sqrt{\left(4 x^2+1\right)}\right)\left(\sqrt{\left(4 x^2+100 x+1\right)}+\sqrt{\left(4 x^2+1\right)}\right)}{\left(\sqrt{\left(4 x^2+100 x+1\right)}+\sqrt{\left(4 x^2+1\right)}\right)} \\ = & \lim _{x \rightarrow \infty} \frac{\left(\left(4 x^2+100 x+1\right)-\left(4 x^2+1\right)\right)}{\left(\sqrt{\left(4 x^2+100 x+1\right)}+\sqrt{\left(4 x^2+1\right)}\right)} \\ = & \lim _{x \rightarrow \infty} \frac{100 x}{\left(\sqrt{\left(4 x^2+100 x+1\right)}+\sqrt{\left(4 x^2+1\right)}\right)} \end{aligned}

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

\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{100}{\left(\sqrt{\left(4+\frac{100}{x}+\frac{1}{x^2}\right)}+\sqrt{\left.\left(4+\frac{1}{x^2}\right)\right)}\right.} \\ & =\frac{100}{4} \\ & =25 \end{aligned}

Posted by

Devendra Khairwa

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