Question

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

Answer 1

The limit tends to `oo`.

Explanation:

We have to solve `lim_(xrarr2^+)(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`:

`(2.1+5)/(2.1-2)=7.1/0.1=71`

For `2.01`:

`(2.01+5)/(2.01-2)=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_(xrarr2)(x+5)/(x-2)=+-oo`

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 `xrarr2^+`, `-oo` is not valid.

So we say `lim_(xrarr2^+)(x+5)/(x-2)=oo`