# How do you determine the limit of (x+5)/(x-2) as x approaches 2+?

Mar 3, 2018

The limit tends to $\infty$.

#### Explanation:

We have to solve ${\lim}_{x \rightarrow {2}^{+}} \frac{x + 5}{x - 2}$

Directly inputting will not work, as the denominator will give $0$, and division by $0$ is not possible.

So we make $x$ get closer and closer to $2$.
For example, we take $2.1$, then $2.01$, $2.001$, and so on.

For $2.1$:

$\frac{2.1 + 5}{2.1 - 2} = \frac{7.1}{0.1} = 71$

For $2.01$:

$\frac{2.01 + 5}{2.01 - 2} = \frac{7.01}{0.01} = 701$

And so on.

We see a pattern. The closer we get to $x = 2$, the value of the limit increases dramatically.

This is because the division of a number by an extremely small one gives us an extremely large answer.

Therefore, we can say:

${\lim}_{x \rightarrow 2} \frac{x + 5}{x - 2} = \pm \infty$

However, the questions asks us to get to two from values larger than it, as shown in the examples. This gives us a positive denominator and numerator, showing us that, when $x \rightarrow {2}^{+}$, $- \infty$ is not valid.

So we say ${\lim}_{x \rightarrow {2}^{+}} \frac{x + 5}{x - 2} = \infty$