# Thread: problems of limit at infinity

1. ## problems of limit at infinity

Dear all,

I have two similar questions but I am not getting the idea how to solve it the second one, is the approach in first part is correct or not???

1. $\mathop {\lim }\limits_{x \to + \infty } {\left( {x + \frac{2}{x}} \right)^{3x}}$
Let $y = {\left( {x + \frac{2}{x}} \right)^{3x}}$, then
$\mathop {\lim }\limits_{x \to + \infty } y = \mathop {\lim }\limits_{x \to + \infty } {e^{\ln {{\left( {x + \frac{2}{x}} \right)}^{3x}}}}$

$= {e^{3x.\ln \left( {x + \frac{2}{x}} \right)}} = {e^{\mathop {\lim }\limits_{x \to + \infty } 3x.\ln \left( {x + \frac{2}{x}} \right)}}$

$= {e^{ + \infty . + \infty }} = + \infty$

Am I right here???

and Second one is
2. $\mathop {\lim }\limits_{x \to - \infty } {\left( {\left| x \right| + \frac{2}{x}} \right)^{3x}}$
how to proceed here???

2. That seems like an awful lot of work.

On #1, the part in the parentheses is x + 2/x. This clearly increases without bound. Unless something is happening to control it, we have divergence. For example, the exponenent might be (1/(3x)) or (-x). If it clearly diverges, don't go to a whole lot of effort.

See if this helps indirectly with #2.