Question

Graph the following function and then find the specified limits. When necessary, state that the limit does not exist.f(x)equals=left brace Start 3 By 2 Matrix 1st Row 1st Column x minus 3 2nd Column if x less than 5 2nd Row 1st Column 2 2nd Column if 5 less than or equals x less than or equals 6 3rd Row 1st Column x plus 4 2nd Column if x greater than 6 EndMatrixx−3 if x<52 if 5≤x≤6x+4 if x>6;findModifyingBelow lim With x right arrow 5limx→5 f(x)andModifyingBelow lim With x right arrow 6limx→6 f(x)

Answer 1

If I'm reading the question right, you have

$f(x)=\begin{cases}x-3&\text{for }x<5\\2&\text{for }5\le x\le6\\x+4&\text{for }x>6\end{cases}$

and you have to find

$\displaystyle\lim_{x\to5}f(x)\text{ and }\lim_{x\to6}f(x)$

The limits exist if the limits from either side exist. We have

$\displaystyle\lim_{x\to5^-}f(x)=\lim_{x\to5}(x-3)=2$

$\displaystyle\lim_{x\to5^+}f(x)=\lim_{x\to5}2=2$

$\implies\displaystyle\lim_{x\to5}f(x)=2$

and

$\displaystyle\lim_{x\to6^-}f(x)=\lim_{x\to6}2=2$

$\displaystyle\lim_{x\to6^+}f(x)=\lim_{x\to6}(x+4)=10$

$\implies\displaystyle\lim_{x\to6}f(x)\text{ does not exist}$