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

For $\lim _ { x \rightarrow 5 ^ { + } } F ( x )$, we approach 5 from x-values larger than 5. When x is larger than 5, the denominator of F(x) is positive, so the function is equal to $\frac{x-5}{x-5}$, which simplifies to 1. Thus, $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) = 1$.

For $\lim _ { x \rightarrow 5 ^ { - } } F ( x )$, we approach 5 from x-values less than 5. When x is less than 5, the denominator of F(x) is negative, so the function is equal to $\frac{x-5}{-(x-5)}$, which simplifies to $-1$. Thus, $\lim _ { x \rightarrow 5 ^ { - } } F ( x ) = -1$.

Since the left and right limits are not equal, the two-sided limit does not exist. Therefore, we do not choose option A. Option B and E have incorrect values for one or both of the limits. Option C has the incorrect value for $\lim _ { x \rightarrow 5 ^ { + } } F ( x )$. Thus, the correct answer choice is D.

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

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