Question #bc838

Nov 17, 2017

We can't, because that is not the limit. If you intended to write ${\lim}_{x \to 10} 3 - \frac{4 x}{5} = - 5$, then see explanation below.

Explanation:

One of the first steps in checking the limit of a function at an x-value is seeing if the function itself is defined for that value, followed by whether it will be defined for nearby x-values.

In this case our function is:

$\frac{3 - 4 x}{5}$

Because this is a simple linear function, it is self evident that the function will be continuous throughout. Thus, the limit shall exist everywhere, and will be equal to the value $f \left({x}_{o}\right)$ for any ${x}_{0}$ in the domain.

Knowing this, we plug in $x = 10$

$\frac{3 - 4 \left(10\right)}{5} = \frac{3 - 40}{5} = - \frac{37}{5} = - 7 \frac{2}{5} = - 7.4$

This is not equal to the $- 5$ you sought, ergo, $- 5$ is not the appropriate limit.

It is possible that you mis-wrote the function, but for the function as you have written it, you cannot prove $- 5$ is the limit because it is not the limit.

If you instead meant to write:

$3 - \left(4 \frac{x}{5}\right)$

You can perform the same process as above. Again, since the function is linear and continuous throughout, the function will be equal to its own limit at any point.

$f \left(10\right) = 3 - \frac{4 \left(10\right)}{5} = 3 - \frac{40}{5} = 3 - 8 = - 5$