Question

lim f(x)=4 and lim f(x)=4, but f(6)= -4.
x→6^- x→6^+
what can you say about lim f(x)?
x→6
lim f(x)
x→6
○a. is ∞
○b. is 4
○c. is -4
○d. does not exist

Explanation:

Step1: Recall the limit existence theorem

The limit of a function \( \lim_{x
ightarrow a}f(x) \) exists if and only if \( \lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x) \).

Step2: Apply the theorem to the given problem

Here \( a = 6 \), \( \lim_{x
ightarrow 6^{-}}f(x)=4 \) and \( \lim_{x
ightarrow 6^{+}}f(x)=4 \). Since \( \lim_{x
ightarrow 6^{-}}f(x)=\lim_{x
ightarrow 6^{+}}f(x) = 4 \), by the limit existence theorem, \( \lim_{x
ightarrow 6}f(x) \) exists.
The value of the function \( f(6)=-4 \) does not affect the limit value. The limit as \( x \) approaches \( 6 \) depends on the behavior of the function near \( x = 6 \), not the value of the function at \( x = 6 \).

Answer:

B. is 4