## Evaluating Limits

### Evaluating Limits: Problems and Solutions

You probably already understand the basics of what limits are, and how to find one by looking at the graph of a function. So we’re going to jump right into where most students initially have some trouble: how to actually evaluate or *compute* a limit in homework and exam problems, especially in cases where you initially get 0 divided by 0. We’ll illustrate the six tactics you *must* know, and then let you practice each.

For a discussion of

*why*these approaches work, please read our blog post 0 Divided by 0: Solve Limit Problems in Calculus, Part 1.

*x*into

*f(x)*. If you obtain a number (and in particular, if you don’t get $\dfrac{0}{0}$), you have your answer and are finished. In that case, these problems are completely straightforward.

**Example**.

Find $\displaystyle{\lim_{x \to 2}(x^3-5x + 7)}$.

*Solution*.

\begin{align*} \lim_{x \to 2}(x^3-5x + 7) &= (2)^3 -5(2) + 7 \\ &= 8 -10 + 7 = 5 \quad \cmark \end{align*}

Practice this (simple!) tactic in the next few problems. The solutions are immediately available using the

**Click to View Calculus Solution**toggle. Substitution practice problem #3 illustrates an important point.

As the preceding problem illustrates, when you substitute the limit-value into the function, if you obtain \(\frac{\text{a non-zero number}}{0}\text{,} \), then you know immediately that the limit does not exist (DNE):

\[\text{If} \quad \lim_{x \to a}f(x) = \frac{\text{a non-zero number}}{0}\] \[\text{then the limit does not exist (DNE).}\]

**Example**.

Find $\displaystyle{\lim_{x \to 2}\frac{x^2-4}{x-2}}$.

*Solution.*

We first try substitution:

\begin{align*}

\lim_{x \to 2}\frac{x^2-4}{x-2} &= \frac{2^2 -4}{2 – 2} = \frac{0}{0}

\end{align*}

Because this limit is in the form of $\dfrac{0}{0}$, it is “indeterminate”—we don’t yet know what it is.

So let’s factor the numerator:

$\displaystyle{\lim_{x \to 2}\frac{x^2-4}{x-2}}$ | \(=\) | $\displaystyle{\lim_{x \to 2}\frac{(x+2)(x-2)}{x-2}}$ | ||||

Ah, now we can cancel the problematic term: | ||||||

\(=\) | $\displaystyle{\lim_{x \to 2}\frac{(x+2)\cancel{(x-2)}}{\cancel{x-2}}}$ | |||||

\(=\) | $\displaystyle{\lim_{x \to 2} \,(x+2)}$ | |||||

And now easy Substitution to finish: | ||||||

\(=\) | $\displaystyle{2 + 2 = 4 \quad \cmark}$ |

Practice the tactic of factoring to find the limit in the next few problems. These are straightforward once you learn to recognize what to do.

*Note*: Every Calculus exam on limits that we’ve ever seen has at *least* one problem that requires this tactic.

**Example**.

Find $\displaystyle{ \lim_{x \to 0}\dfrac{\sqrt{x+5} – \sqrt{5}}{x}}$.

*Solution*.

As always, we first try Substitution:

$$\lim_{x \to 0}\dfrac{\sqrt{x+5} – \sqrt{5}}{x} = \frac{\sqrt{5} – \sqrt{5}}{0} = \dfrac{0}{0}$$

Since this limit is in the form of $\dfrac{0}{0}$, it is indeterminate—we don’t yet know what it is. We have more work to do.

So let’s try to rationalize the expression. The square-root part is $\sqrt{x+5} – \sqrt{5}$, so we multiply both the numerator and the denominator by the conjugate $\dfrac{\sqrt{x+5} + \sqrt{5}}{\sqrt{x+5} + \sqrt{5}}=1$:

\begin{align*}

\lim_{x \to 0}\dfrac{\sqrt{x+5} – \sqrt{5}}{x} &= \lim_{x \to 0}\dfrac{\sqrt{x+5} – \sqrt{5}}{x} \cdot \dfrac{\sqrt{x+5} + \sqrt{5}}{\sqrt{x+5} + \sqrt{5}} \\[8px]
&= \lim_{x \to 0}\dfrac{\sqrt{x+5}\sqrt{x+5} + \sqrt{x+5}\sqrt{5} – \sqrt{5}\sqrt{x+5} -\sqrt{5}\sqrt{5}}{x[\sqrt{x+5} + \sqrt{5}]} \\[8px]
&= \lim_{x \to 0}\dfrac{\sqrt{x+5}\sqrt{x+5} + \cancel{\sqrt{x+5}\sqrt{5}} – \cancel{\sqrt{5}\sqrt{x+5}} -\sqrt{5}\sqrt{5}}{x[\sqrt{x+5} + \sqrt{5}]} \\[8px]
&= \lim_{x \to 0}\dfrac{(x+5) – 5}{x[\sqrt{x+5} + \sqrt{5}]} \\[8px]
&= \lim_{x \to 0}\dfrac{x}{x[\sqrt{x+5} + \sqrt{5}]} \\[8px]
&= \lim_{x \to 0}\dfrac{\cancel{x}}{\cancel{x}[\sqrt{x+5} + \sqrt{5}]} \\[8px]
&= \lim_{x \to 0}\dfrac{1}{\sqrt{x+5} + \sqrt{5}} \\[8px]
&= \dfrac{1}{\sqrt{0+5} + \sqrt{5}} = \dfrac{1}{2\sqrt{5}} \quad \cmark

\end{align*}

Notice that when we multiplied by the conjugate, we multiplied out all of the terms in the numerator, because that’s how we get rid of the square root. But we *didn’t* multiply the terms in the denominator; instead we kept writing it as $x \left(\sqrt{x+5} + \sqrt{5} \right)$. That’s because a few steps later the *x* canceled.

Something similar will always happen, so in that early step don’t multiply out the part that you didn’t set out to rationalize. Instead just carry those terms along for a while, until you can cancel something.

1. Remember your fundamental trig identities. For instance:

- $\tan x = \dfrac{\sin x}{\cos x}$
- $\csc x = \dfrac{1}{\sin x}$
- $\sin^2(x) + \cos^2(x) = 1$

2. Early in the semester, there are two “Special Limits” you just have to memorize:

\begin{align*}

\text{I. } & \lim_{x \to 0}\frac{\sin(x)}{x} = 1 \\[16px]
\text{II. } & \lim_{x \to 0}\frac{1-\cos(x)}{x} = 0

\end{align*}

The following problems illustrate how you’ll frequently have to make use of these facts.

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