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What can be said about limits that have the form nonzero over zero?

Let’s cut to the chase:

Which of the following limits are of the form $\relax \boldsymbol {\tfrac {\#}{0}}$?
$\lim _{x\to -1} \frac {1}{(x+1)^2}$ $\lim _{x\to 2}\frac {x^2-3x+2}{x-2}$ $\lim _{x\to 0}\frac {\sin (x)}{x}$ $\lim _{x\to 2}\frac {x^2-3x-2}{x-2}$ $\lim _{x\to 1}\frac {e^x}{\ln (x)}$

In our next example, let’s see what is going on with limits of the form $\relax \boldsymbol {\tfrac {\#}{0}}$.

We are now ready for our next definition.

Note: Saying “the limit is equal to infinity” describes more precisely the behavior of the function $f$ near $a$, than just saying ”the limit does not exist”.

Let’s consider a few more examples.

Here is our final example.

Some people worry that the mathematicians are passing into mysticism when we talk about infinity and negative infinity. However, when we write all we mean is that as $x$ approaches $a$, $f(x)$ becomes arbitrarily large and $|g(x)|$ becomes arbitrarily large, with $g(x)$ taking negative values.