1. ## slant asymptotes

can someone help me find the slant asymptote of y=(e^x)-x...please and thanks

2. If a line of the form $\displaystyle mx + b$ and $\displaystyle \lim_{x \to \infty} \left[ f(x) - (mx + b)\right] = 0$, then your line is a slant asymptote.

So we want: $\displaystyle \lim_{x \to \infty} \left[ e^x - x -( mx + b)\right] = 0$

We know $\displaystyle \lim_{x \to \infty} {\color{magenta}e^{x} = 0}$ and to make things simpler, consider $\displaystyle \color{blue}b=0$

So we want: $\displaystyle \lim_{x \to \infty} \left[ {\color{magenta}e^{x}} - x -( mx + {\color{blue}b})\right] = \lim_{x \to \infty} \left( -x - mx\right) = 0$

So for what values of m would $\displaystyle \color{red} -x - mx = 0$ (since $\displaystyle \lim_{x \to \infty} 0 = 0$)?

Once we find that, we can piece it all together: $\displaystyle \lim_{x \to \infty}e^x - x - (mx + b) = \lim_{x \to \infty} {\color{magenta}e^x} {\color{red}- x - mx} - {\color{blue}b} = \lim_{x \to \infty} {\color{magenta}0} + {\color{red}0} + {\color{blue}0} = 0$

3. Originally Posted by johntuan
can someone help me find the slant asymptote of y=(e^x)-x...please and thanks
$\displaystyle y = e^x - x$ approaches the line $\displaystyle y = - x$ as $\displaystyle x \rightarrow - \infty$.