Limit of an indeterminate form $\infty - \infty$.

Question

I would appreciate some help figuring out a way to solve the following limit: $$\lim_{x\rightarrow \infty} \left(x-e^x \right)$$

I know that $e^x$ is much larger than $x$, and therefore the limit will be $-\infty$. What I would like to know is whether or not there is a way to solve this algebraically. Any one able to help? I can't seem to force L'Hopital's Rule here, nor was I able to use the limit definition of $e$.

Thanks, Mada

Answer 1

You can do some rearranging: $$\lim_{x\rightarrow \infty}x-e^x=\lim_{x\rightarrow \infty}xe^{-x}/e^{-x}+1/e^{-x}=\lim_{x\rightarrow \infty}\frac{x/e^{x}-1}{e^{-x}}=\lim_{x\rightarrow \infty}-e^{x}=-\infty$$ L'hopitals is used implicity here with $\lim_{x\rightarrow \infty}\frac{x}{e^x}$.