# Find the limit ?

Mar 18, 2017

I would say that it tends to $+ \infty$ from the left and $- \infty$ from the right, or the limit diverges.

#### Explanation:

If you use $x = - 5$ into your function you get:
$\frac{25 + 35 + 10}{25 - 25} = \frac{70}{0}$
this tells us that as $x \to - 5$ our function will become very big, so our limit would tend to $\pm \infty$ depending upon the side you are considering:
Graphically:
graph{(x^2-7x+10)/(x^2-25) [-12.66, 12.65, -6.33, 6.33]}
as you can see at $x = - 5$ you have a discontinuity,

Mar 18, 2017

${\lim}_{x \to - 5} \left(\frac{{x}^{2} - 7 x + 10}{{x}^{2} - 25}\right) \text{ does not exist}$

#### Explanation:

${\lim}_{x \to - 5} \left(\frac{{x}^{2} - 7 x + 10}{{x}^{2} - 25}\right)$

Use direct substitution:
$\frac{{\left(- 5\right)}^{2} - 7 \left(- 5\right) + 10}{{\left(- 5\right)}^{2} - 25} = \frac{70}{0}$

Thus, the limit does not exist because $\frac{70}{0}$ is undefined.

Check with a graph:
graph{(x^2-7x+10)/(x^2-25) [-12.66, 12.65, -6.33, 6.33]}
As you can see, at $x = - 5$, the function approaches $\infty$ from the left and $- \infty$ from the right.

*Note: We cannot use L'Hospital's rule because $\frac{70}{0}$ is not an indeterminate case