Question

Evaluate.
\[\lim_{x \rightarrow -\infty} \left[ \frac{ 2x }{ \sqrt{4x^2+32x}+x } \right]\]

Answer 1

\(\large \color{black}{\begin{align}\lim_{x \rightarrow -\infty} \left[ \frac{ 2x }{ \sqrt{4x^2+32x}+x } \right] \hspace{.33em}\\~\\
\implies \lim_{x \rightarrow -\infty} \left[ \frac{ 2 }{ \sqrt{4+\dfrac{32}{x}}+1 } \right] \hspace{.33em}\\~\\
\end{align}}\)

Answer 2

\(\lim_{x \rightarrow -\infty} \left[ \frac{ 2x }{ \sqrt{4x^2+32x}+x } \right]=\lim_{x \rightarrow -\infty} \left[ \frac{ \frac{1}{x}2x }{\frac{1}{x}( \sqrt{4x^2+32x}+x )} \right]=\lim_{x \rightarrow -\infty} \left[ \frac{ 2 }{ \frac{1}{x}\sqrt{4x^2+32x}+1 } \right]\\=\lim_{x \rightarrow -\infty} \left[ \frac{ 2 }{ \sqrt{\frac{4x^2}{x^2}+\frac{32}{x}}+1 } \right]=\lim_{x \rightarrow -\infty} \left[ \frac{ 2 }{ \sqrt{4+\frac{32}{x}}+1 } \right]\)

Now run the limit

Answer 3

okay...

Answer 4

So do you understand what is going on?

Answer 5

I think so... the answer is 2/3? @geerky42

Answer 6

Yeah

Answer 7

Thanks!