Question

Evaluate the Limit?

`lim_(xrarr oo) [x(tan(2/x))]`

Answer 1

The limit is `2`

Explanation:

We can rewrite as

`L = lim_(x-> oo) tan(2/x)/(1/x)`

We see that `tan(2/x)` converges to `0` as `x-> oo` because the larger the value of `x` the smaller the expression within the tangent becomes and the closer `tan(2/x)` becomes to `0`. Also, the limit `lim_(x-> oo) 1/x = 0` is commonly used. Therefore, we may use l;Hosptial's rule.

`L = lim_(x->oo) (-2/x^2 * sec^2(2/x))/(-1/x^2)`

`L = lim_(x-> oo) 2sec^2(2/x)`

The same principal applies with `2/x` as `1/x`: the limit as `x` approaches infinity always remains `0`.

`L = 2sec^2(0)`

`L = 2(1)`

`L = 2`

A graphical verification confirms.

In the above graph, the red curve is `xtan(2/x)` and the blue line is `y = 2`. As you can see, the curve converges onto the line.

Hopefully this helps!