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We want to evaluate limits where the Limit Laws do not directly apply.

In the last section, we were interested in the limits we could compute using continuity and the limit laws. What about limits that cannot be directly computed using these methods? Let’s think about an example. Consider Here in light of this, you may think that the limit is one or zero. Not so fast. This limit is of an indeterminate form. What does this mean? Read on, young mathematician.

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

Let’s consider an example with the function above:

Let’s consider some more examples of the form $\relax \boldsymbol {\tfrac {0}{0}}$.

Finally, we’ll look at one more example.

All of the examples in this section are limits of the form $\relax \boldsymbol {\tfrac {0}{0}}$. When you come across a limit of the form $\relax \boldsymbol {\tfrac {0}{0}}$, you should try to use algebraic techniques to come up with a continuous function whose limit you can evaluate.

Notice that we solved multiple examples of limits of the form $\relax \boldsymbol {\tfrac {0}{0}}$ and we got different answers each time. This tells us that just knowing that the form of the limit is $\relax \boldsymbol {\tfrac {0}{0}}$ is not enough to compute the limit. The moral of the story is

Limits of the form $\relax \boldsymbol {\tfrac {0}{0}}$ can take any value.

Finally, you may find it distressing that we introduced a form, namely $\relax \boldsymbol {\tfrac {0}{0}}$, only to end up saying they give no information on the value of the limit. But this is precisely what makes indeterminate forms interesting… they’re a mystery!