함수의 그래프 그리기

(a) The value of the numerator at $x=c$ is $e^{-c} \ne 0$, while the value of the denominator is $0$ at $x=c$. Thus, $\lim_{x \to c^+} \frac{e^{-x}}{x-c} = \infty$ and $\lim_{x \to c^-} \frac{e^{-x}}{x-c} = -\infty$. This makes the line $x=c$ a vertical asymptote for $q$. (b) We can write $q(x)$ as $q(x) = \frac{1}{(x-c)e^x}$ and thus see the denominator increases without bound as $x \to \infty$. Therefore, $\lim_{x \to \infty} \frac{e^{-x}}{x-c} = 0$. This makes the line $y=0$ a horizontal asymptote for $q$. As $x \to -\infty$, $e^{-x}$ increases without bound, while $(x-c)$ decreases without bound. Thus, $q$ has an indeterminate form as $x \to -infty$. Applying L'Hopital's Rule gives us $$\lim_{x \to -\infty} \frac{e^{-x}}{x-c} = \lim_{x \to -\infty} \frac{-e^{-x}}{1} = -\infty\,.$$ (c) Using $q(x) = \frac{1}{(x-c)e^x}$, the quotient and product rules along with algebraic simplification show that $$q'(x) = -\frac{(x-c)e^x + e^x}{(x-c)^2\left(e^x\right)^2} = -\frac{x-c+1}{(x-c)^2e^x}\,.$$ The function $q$ has critical points in its domain where $q'(x) = 0$ or $q'(x)$ doesn't exist. Note that $q'(x)$ is undefined at $x=c$, but $c$ is not in the domain of mq. Thus, the only critical points of $q$ occur when $q'(x)=0$. The only way the fraction $q'(x)$ can be $0$ is if the numerator is $0$. Since $x-c+1=0$ when $x=c-1$, the only critical number for $q$ is $x=c-1$. (d) The values of the exponential function $e^x$ and the quadratic $(x-c)^2$ are positive for $x \neq c$, so the sign of $q'(x) = -\frac{x-c+1}{(x-c)^2e^x}$ depends entirely on the sign of $x-(c-1)$. When $x < c-1$ we see that $q'(x) > 0$ and when $x > c-1$ we have $q'(x) < 0$. Thus, $q$ is increasing on $(-\infty, c-1)$ and is decreasing on $(c-1,c)$, so it follows that $q(c-1)$ is a relative maximum value. This value is a local maximum, since the value of $q$ can be made as large as we like as shown in part (b). (e) We've shown that a typical member of the family of functions given by $q(x) = \frac{1}{(x-c)e^x}$ has a vertical asymptote at $x = c$ and a critical number at which a local maximum occurs one unit to the left of $c$ at $x = c-1$. A typical such graph is shown below. Note how $q$ is increasing and decreasing according to our earlier work in (d).