Limit of the derivative of an expression

Question

The expression is : $\theta(t)$ = tan^-1 (t/500)

The derivative is(unless I'm wrong): $\theta'(t) = 1/(1+(t^2/250000))$

is that correct?

Then I'm asked to find the limit of $\theta'(t)$ when t = +-$\infty$

Is this then the correct appraoch?

$\theta'(t)$ when t = +-$\infty$

= 1/(1+$\infty$)

= 1/$\infty$

= 0.

I was first thinking of applying l'hopitals rule, but I guess since the expression is not 0/0, or $\infty$/$\infty$, that's not the way to do it?

Answer 1

Your answer is almost correct except the expression of $\theta'(t)$. There is a coefficient of $t$ in $\theta(t)$, so it should be $$ \theta'(t)=\mathbf{\frac{1}{500}}\cdot\frac{1}{1+t^2/250000} $$ However, it doesn't affect the result. When $t$ goes to infinity, $\theta'(t)$ does go to $0$ in the way as you said.

I was first thinking of applying l'hopitals rule, but I guess since the expression is not $0/0$, or $\infty / \infty$, that's not the way to do it?

You're correct here.