Question

the graph of the rational function f is shown with a hole at (1,1). which of the following statements about the limits below are true? lim_{x→1^{-}}f(x) and lim_{x→1^{+}}f(x) a. both limits are defined, but the values are unequal so the point is not defined. b. both limits are defined, so the limit as x approaches 1 is defined and lim_{x→1}f(x)=1 c. neither limits are defined, because the graph has a hole at 1. d. neither limits are defined, but the point at 1 is defined.

Explanation:

Step1: Recall limit definition

The limit of a function as $x$ approaches a value exists if the left - hand limit $\lim_{x
ightarrow a^{-}}f(x)$ and the right - hand limit $\lim_{x
ightarrow a^{+}}f(x)$ are equal. A hole in the graph of a function at $x = a$ does not prevent the limit from existing at $x=a$.

Step2: Analyze left - hand and right - hand limits

For a rational function $f(x)$ with a hole at $(1,1)$, the left - hand limit $\lim_{x
ightarrow 1^{-}}f(x)$ and the right - hand limit $\lim_{x
ightarrow 1^{+}}f(x)$ can be equal. Just because there is a hole at $x = 1$ (i.e., the function is not defined at $x = 1$ in its original form), the limits $\lim_{x
ightarrow 1^{-}}f(x)$ and $\lim_{x
ightarrow 1^{+}}f(x)$ can still be well - defined. If $\lim_{x
ightarrow 1^{-}}f(x)=\lim_{x
ightarrow 1^{+}}f(x)=L$, then $\lim_{x
ightarrow 1}f(x)=L$.

Step3: Evaluate each option

• Option A: Both limits are defined, but the values are unequal so the point is not defined. This is incorrect because if the left - hand and right - hand limits are unequal, the limit as $x$ approaches 1 does not exist.
• Option B: Both limits are defined, so the limit as $x$ approaches 1 is defined. This is correct. When the left - hand and right - hand limits are equal (even if the function is not defined at the point), the limit exists.
• Option C: Neither limits are defined, because the graph has a hole at 1. This is incorrect. A hole does not mean the limits do not exist.
• Option D: Neither limits are defined, but the point at 1 is defined. This is incorrect. A hole implies the function is not defined at that point, and the limits can be defined.

Answer:

B. Both limits are defined, so the limit as $x$ approaches 1 is defined.