Question

HELP!!! I DONT UNDERSTAND

Answer 1

Notice that both pieces of $f(x)$ are continuous on their respective domains. That is, $e^{6x}$ exists and is defined for all $x\le0$; the same goes for $9-8\cos x$ for all $x>0$.

a. Because of the continuity of each piece, we have

$\lim_{x\to-1}f(x)=f(-1)=e^6$

where we take $f(x)=e^{6x}$ because that's how $f$ is defined if $x\le0$.

b. For $f$ to be continuous at $x=0$, both one-sided limits have to come out to be the same number, $f(0)$. By definition of

The limits are

$\displaystyle\lim_{x\to0^-}f(x)=\lim_{x\to0}e^{6x}=1$

$\displaystyle\lim_{x\to0^+}f(x)=\lim_{x\to0^+}(9-8\cos x)=9-8=1$

c. $x\to-\infty$ means $x$ is negative, for which $f(x)=e^{6x}$. Then

$\displaystyle\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}e^{6x}=0$