Stuck on a homework question: if $t = \frac{1}{x}$ 
If $\displaystyle t = \frac{1}{x}$ then
a) Explain why $\displaystyle\lim_{x \to 0^-}f(x)$ is equivalent to $\displaystyle\lim_{t \to -\infty}f\left(\frac{1}{t}\right)$
b) Using that, rewrite $\displaystyle\lim_{x \to 0^-} e^\frac{1}{x}$ as equivalent limit involving $t$ only. Evaluate the corresponding limit.
c) Use a similar technique to evaluate $\displaystyle\lim_{x \to \pm\infty} \ln\left(\frac{2}{x^3}\right)$

On a, I thought that they are equivalent because $f(x)$ is the inverse of $t$, is that correct?
On b, I get $\displaystyle\lim_{x \to 0^-} e^t$. How would I solve this limit? Isn't it the $t$'th root of $e$? Im confused...
And c just confuses me to begin with. How would I calculate that limit? I don't know where to start on that one and just need some help with getting my mind around it.
Thanks
 A: (a) does not have to do with $f$; rather, the point is that if $t=\frac{1}{x}$, then $x=\frac{1}{t}$, and $\lim\limits_{t\to-\infty}x = 0^-$ (that is, as $t$ approaches $-\infty$, the value of $\frac{1}{t}$ approaches $0$ from the left; and conversely, as $\frac{1}{t}=x$ approaches $0$ from the left, the value of $t=\frac{1}{x}$ approaches $-\infty$. So both limits are considering arguments that approach the same thing.
For (b), you forgot to change the limit as well: your limit is still a limit of $x$, not a limit of $t$. Notice that in (a), both the function and the argument (the stuff under the "$\lim$") change. So rather than
$$\lim_{x\to 0^-}e^t$$
you should have a limit as $t\to-\infty$:
$$\lim_{x\to 0^-}e^{\frac{1}{x}} = \lim_{t\to-\infty}e^{t}.$$
As to evaluating this limit, it should follow from what you know about the exponential function. This is a standard limit for the exponential.
For (c), use "a similar technique": set $t = \frac{1}{x^3}$. As $x\to\infty$, what happens to $t$? It approaches $0$. From what side? From the positive side. So
$$\lim_{x\to\infty}\ln\left(\frac{2}{x^3}\right) = \lim_{x\to\infty}\ln\left(2\left(\frac{1}{x^3}\right)\right) = \lim_{t\to 0^+}\ln\left(2t\right).$$
Can you evaluate that limit?
Similarly for the limit as $x\to-\infty$. If $t=\frac{1}{x^3}$, what happens to $t$ as $x\to-\infty$?
A: (a) $f(x)$ is the inverse of $t$ ... incorrect   
(b) $x \to 0^+$ does not convert to $t \to 0^+$
(c)  First do (a) and (b), then use the insight you gain to do (c).  Note: you gain insight when you do them yourself, not when others do them for you...
