Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$. Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) , \lim _ { x \rightarrow 5 ^ { - } } F ( x )$
A) both 1
B) 2 and 1
C) 2 and $- 1$
D) 1 and $- 1$
E) both $- 1$

Question

Quizplus · Accepted Answer

When $x>5$, $|x-5|=x-5$ so $F(x)=\frac{x-5}{x-5}=1$. Therefore, $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) =1$.
When $x<5$, $|x-5|=-(x-5)=5-x$ so $F(x)=\frac{x-5}{5-x}=-1$. Therefore, $\lim _ { x \rightarrow 5 ^ { - } } F ( x ) =-1$.
Thus, the answer is D) 1 and $-1$.

Let F(x)=x−5∣x−5∣F ( x ) = \frac { x - 5 } { | x - 5 | }F(x)=∣x−5∣x−5​

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$