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  • Find value of limit:<img alt="\lim _{x \rightarrow \infty}\left(\sqrt{\left(7 x^2+5 x+3\right)}-\sqrt{\left(7 x^2+3\right)}\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7

Find value of limit:\lim _{x \rightarrow \infty}\left(\sqrt{\left(7 x^2+5 x+3\right)}-\sqrt{\left(7 x^2+3\right)}\right)

Option: 1

\frac{5}{2\sqrt{7}}


Option: 2

\frac{5}{\sqrt{7}}


Option: 3

\frac{3}{2\sqrt{7}}


Option: 4

\frac{3}{\sqrt{7}}


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(7 x^2+5 x+3\right)}-\sqrt{\left(7 x^2+3\right)}\right) is \lim _{x \rightarrow \infty}\left(\sqrt{\left(7 x^2+5 x+3\right)}+\sqrt{\left(7 x^2+3\right)}\right)

Hence,

\begin{aligned} & \lim _{x \rightarrow \infty}\left(\sqrt{\left(7 x^2+5 x+3\right)}+\sqrt{\left(7 x^2+3\right)}\right) \\ = & \lim _{x \rightarrow \infty} \frac{\left(\sqrt{\left(7 x^2+5 x+3\right)}-\sqrt{\left(7 x^2+3\right)}\right)\left(\sqrt{\left(7 x^2+5 x+3\right)}+\sqrt{\left(7 x^2+3\right)}\right)}{\left(\sqrt{\left(7 x^2+5 x+3\right)}+\sqrt{\left(7 x^2+3\right)}\right)} \\ = & \lim _{x \rightarrow \infty} \frac{\left(\left(7 x^2+5 x+3\right)-\left(7 x^2+3\right)\right)}{\left(\sqrt{\left(7 x^2+5 x+3\right)}+\sqrt{\left(7 x^2+3\right)}\right)} \\ = & \lim _{x \rightarrow \infty} \frac{5 x}{\left(\sqrt{\left(7 x^2+5 x+3\right)}+\sqrt{\left(7 x^2+3\right)}\right)} \end{aligned}

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

\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{5}{\left(\sqrt{\left(7+\frac{5}{x}+\frac{3}{x^2}\right)}+\sqrt{\left.\left(7+\frac{3}{x^2}\right)\right)}\right.} \\ & =\frac{5}{2 \sqrt{7}} \end{aligned}

Posted by

Rakesh

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