# Determine the limit. \lim_{x \rightarrow 5} \frac{4x}{x-5}

## Question:

Determine the limit.

{eq}\displaystyle \lim_{x \rightarrow 5} \frac{4x}{x - 5} {/eq}

## Limits:

Let {eq}f {/eq} be a function of a real variable {eq}x {/eq}. Let {eq}c, l {/eq} be two fixed numbers. If {eq}f(x) {/eq} approaches the value {eq}l {/eq} as {eq}x {/eq} approaches {eq}c {/eq}, we say {eq}l {/eq} is the limit of the function {eq}f(x) {/eq} as {eq}x {/eq} tends to {eq}c {/eq}. This is written as {eq}\displaystyle \lim_{x\to c} f(x) = l {/eq}. The given limit could be simplified by the substitution of the given value of the variable.

{eq}\begin{align*} \lim_{x\to 5} \frac{4x}{x-5}&=\frac{20}{0}\\ &=\infty \end{align*} {/eq}

Therefore the given limit does not exist.