Question

Find the indicated limit, if it exists. (2 points) limit of f of x as x approaches 5 where f of x equals 5 minus x when x is less than 5, 8 when x equals 5, and x plus 3 when x is greater than 5

Answer 1

It looks like we have

`f(x)=\begin{cases}5-x&\text{for }x<5\\8&\text{for }x=5\\x+3&\text{for }x>5\end{cases}`

and we want to find
`\lim\limits_(x\to5)f(x)`.

Since
`x` is approaching 5, we don't care about the value of
`f(x)` when
`x=5`.

We do care about how
`f(x)` behaves to either side of
`x=5`. If
`x\to5` from below, then
`f(x)=5-x`, so that

`\displaystyle\lim_(x\to5^-)f(x)=\lim_(x\to5)(5-x)=5-5=0`

On the other hand, if
`x\to5` from above, then
`f(x)=x+3`, so that

`\displaystyle\lim_(x\to5^+)f(x)=\lim_(x\to5)(x+3)=5+3=8`

The one-sided limits do not match, since 0 ≠ 8, so the limit does not exist.