Do we need to apply the L'Hôpital's rule to all the polynomial when evaluating infinity?

Question

I have this:

$$-\frac{x}{e^{\frac{x}{10}}}-\frac{10}{e^{\frac{x}{10}}}$$

Do I need to derivate the numerator and denominator of all this expression or just :

$$-\frac{x}{e^{\frac{x}{10}}}$$

And

$$-\frac{10}{e^{\frac{x}{10}}}$$

Keeps the same?

Thanks.

Answer 1

There are several ways to solve this. Here is one:

$$\begin{align*}\lim_{x \to \infty} \left( -\dfrac{x}{ e^{ \tfrac{x}{10} } } - \dfrac{ 10 }{ e^{ \tfrac{x}{10} } } \right) & = \lim_{ x \to \infty} -\dfrac{x+10}{e^{\tfrac{x}{10}}} \\ & \underset{\infty}{\overset{\infty}{=}} -\lim_{x \to \infty} \dfrac{1}{\dfrac{1}{10}\cdot e^{\tfrac{x}{10}}} = 0\end{align*}$$

Answer 2

Note that as $x\to \infty$ the second fraction goes to zero and the L'Hospital's rule does not apply to it.

Thus you just need to work on the first fraction.