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Math Help - slant asymptotes

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    slant asymptotes

    can someone help me find the slant asymptote of y=(e^x)-x...please and thanks
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    If a line of the form mx + b and \lim_{x \to \infty} \left[ f(x) - (mx + b)\right] = 0, then your line is a slant asymptote.

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

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

    So we want: \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 \color{red} -x - mx = 0 (since \lim_{x \to \infty} 0 = 0)?

    Once we find that, we can piece it all together: \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
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    Quote Originally Posted by johntuan View Post
    can someone help me find the slant asymptote of y=(e^x)-x...please and thanks
    y = e^x - x approaches the line y = - x as x \rightarrow - \infty.
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