Math / Calculus

QuestionPoints: 0 of 1 a. $\lim _{x \rightarrow 2} g(x)$ b. $\lim _{x \rightarrow 3} g(x)$ c. $\lim _{x \rightarrow 4} g(x)$ d. $\lim _{x \rightarrow 3.5} g(x)$ a. What is $\lim _{x \rightarrow 2} g(x)$ ? Choose the correct answer below and, if necessary, if in the arswer box to complete your choice A. $\lim _{x \rightarrow 2} g(x)=$ $\square$ B. $\lim _{x \rightarrow 2} g(x)$ does not exist

Studdy Solution

STEP 1

1. The function $g(x)$ is defined by the graph provided.
2. The limit $\lim_{{x \to a}} g(x)$ exists if and only if both the left-hand limit $\lim_{{x \to a^-}} g(x)$ and the right-hand limit $\lim_{{x \to a^+}} g(x)$ exist and are equal.

STEP 2

1. Determine the left-hand limit $\lim_{{x \to 2^-}} g(x)$ .
2. Determine the right-hand limit $\lim_{{x \to 2^+}} g(x)$ .
3. Compare the left-hand limit and the right-hand limit to conclude if the two-sided limit $\lim_{{x \to 2}} g(x)$ exists.

STEP 3

Examine the graph as $x$ approaches 2 from the left to find $\lim_{{x \to 2^-}} g(x)$ .
The graph shows a line segment from $(0, -2)$ to $(2, 4)$ , indicating that as $x$ approaches 2 from the left, $g(x)$ approaches 4. $\lim_{{x \to 2^-}} g(x) = 4$

STEP 4

Examine the graph as $x$ approaches 2 from the right to find $\lim_{{x \to 2^+}} g(x)$ .
The graph shows a point at $(2, 4)$ and a line segment starting from $(2, 4)$ to $(3, 1)$ . Therefore, as $x$ approaches 2 from the right, $g(x)$ also approaches 4. $\lim_{{x \to 2^+}} g(x) = 4$

STEP 5

Compare the left-hand limit and the right-hand limit to conclude the two-sided limit $\lim_{{x \to 2}} g(x)$ .
Since both $\lim_{{x \to 2^-}} g(x)$ and $\lim_{{x \to 2^+}} g(x)$ are equal to 4: $\lim_{{x \to 2}} g(x) = 4$
Solution: The correct answer is $\lim_{{x \to 2}} g(x) = 4$

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