Question

find if limit exist ​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\sf{f(x)=\begin{cases}\sf{\dfrac{|x+7|}{x+7}\,\,\,\,\,\,\,,x\ne-7}\\\\\sf{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x=-7}\end{cases}}$

$\tt{LHL:\displaystyle\lim_{x\to(-7)^-}f(x)=\lim_{h\to0}f(-7-h)}$

$\tt{\displaystyle\,=\lim_{h\to0}\dfrac{|-7-h+7|}{-7-h+7}}$

$\tt{\displaystyle\,=\lim_{h\to0}\dfrac{|-h|}{-h}}$

$\tt{\displaystyle\,=-\lim_{h\to0}\dfrac{h}{h}}$

$\tt{\displaystyle\,=-\lim_{h\to0}(1)}$

$\tt{\displaystyle\,=-1}$

$\tt{RHL:\displaystyle\lim_{x\to(-7)^+}f(x)=\lim_{h\to0}f(-7+h)}$

$\tt{\displaystyle\,=\lim_{h\to0}\dfrac{|-7+h+7|}{-7+h+7}}$

$\tt{\displaystyle\,=\lim_{h\to0}\dfrac{|h|}{h}}$

$\tt{\displaystyle\,=\lim_{h\to0}\dfrac{h}{h}}$

$\tt{\displaystyle\,=1}$

Since $LHL\ne\,RHL$

Hence, limit does not exists