# Evaluate the Limit?

## ${\lim}_{x \rightarrow \infty} \left[x \left(\tan \left(\frac{2}{x}\right)\right)\right]$

Mar 20, 2018

The limit is $2$

#### Explanation:

We can rewrite as

$L = {\lim}_{x \to \infty} \tan \frac{\frac{2}{x}}{\frac{1}{x}}$

We see that $\tan \left(\frac{2}{x}\right)$ converges to $0$ as $x \to \infty$ because the larger the value of $x$ the smaller the expression within the tangent becomes and the closer $\tan \left(\frac{2}{x}\right)$ becomes to $0$. Also, the limit ${\lim}_{x \to \infty} \frac{1}{x} = 0$ is commonly used. Therefore, we may use l;Hosptial's rule.

$L = {\lim}_{x \to \infty} \frac{- \frac{2}{x} ^ 2 \cdot {\sec}^{2} \left(\frac{2}{x}\right)}{- \frac{1}{x} ^ 2}$

$L = {\lim}_{x \to \infty} 2 {\sec}^{2} \left(\frac{2}{x}\right)$

The same principal applies with $\frac{2}{x}$ as $\frac{1}{x}$: the limit as $x$ approaches infinity always remains $0$.

$L = 2 {\sec}^{2} \left(0\right)$

$L = 2 \left(1\right)$

$L = 2$

A graphical verification confirms. In the above graph, the red curve is $x \tan \left(\frac{2}{x}\right)$ and the blue line is $y = 2$. As you can see, the curve converges onto the line.

Hopefully this helps!