If $\lim _{x \rightarrow 3^{-}} f(x)=4.1$ , then if $\lim _{x \rightarrow 3} f(x)$ exists, find the value where it exists.
A) $4.1$
B) 1
C) $4 .4$
D) 2
E) 3

Question

Quizplus · Accepted Answer

Since the left-hand limit as $x$ approaches $3$ is $4.1$, we know that as we approach $3$ from the left, $f(x)$ is getting arbitrarily close to $4.1$. If $\lim _{x \rightarrow 3} f(x)$ exists, it must be the case that the right-hand limit as $x$ approaches $3$ is also $4.1$ (since if it were any other value, the limit would not exist). Thus, we know that as we approach $3$ from the right, $f(x)$ is also getting arbitrarily close to $4.1$. Since the limit from both sides is the same, $\lim _{x \rightarrow 3} f(x)$ exists and must be equal to $4.1$. Therefore, the answer is A) $4.1$.

If lim⁡x→3−f(x)=4.1\lim _{x \rightarrow 3^{-}} f(x)=4.1limx→3−​f(x)=4.1 , Then If lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x)limx→3​f(x)

If $\lim _{x \rightarrow 3^{-}} f(x)=4.1$ , Then If $\lim _{x \rightarrow 3} f(x)$