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Math  /  Calculus

QuestionUse the figure below, which gives a graph of the function f(x)f(x), to give values for the indicated limits. If a limit does not exist, enter DNE.
Note: You can click on the graph to enlarge the image. (a) lim⁡x→−2−f(x)=\lim _{x \rightarrow-2^{-}} f(x)= □\square help (limits) (b) lim⁡x→−2+f(x)=\lim _{x \rightarrow-2^{+}} f(x)= □\square (c) lim⁡x→−2f(x)=\lim _{x \rightarrow-2} f(x)= □\square (d) lim⁡x→−1f(x)=\lim _{x \rightarrow-1} f(x)= □\square

Studdy Solution

STEP 1

1. The graph of the function f(x) f(x) shows the behavior of the function as x x approaches specific points.
2. The limits involve analyzing the behavior of f(x) f(x) as x x approaches the given points from both the left and the right.
3. We need to determine the limit values or identify if the limit does not exist (DNE).

STEP 2

1. Determine the left-hand limit as x x approaches −2-2.
2. Determine the right-hand limit as x x approaches −2-2.
3. Determine the limit as x x approaches −2-2 (considering both left and right limits).
4. Determine the limit as x x approaches −1-1.

STEP 3

Identify the left-hand limit of f(x) f(x) as x x approaches −2-2.
From the graph, as x x approaches −2-2 from the left (x→−2− x \to -2^- ), f(x) f(x) approaches 0. Thus, lim⁡x→−2−f(x)=0 \lim_{x \to -2^-} f(x) = 0

STEP 4

Identify the right-hand limit of f(x) f(x) as x x approaches −2-2.
From the graph, as x x approaches −2-2 from the right (x→−2+ x \to -2^+ ), f(x) f(x) approaches -4. Thus, lim⁡x→−2+f(x)=−4 \lim_{x \to -2^+} f(x) = -4

STEP 5

Determine the limit of f(x) f(x) as x x approaches −2-2 considering both left and right limits.
Since the left-hand limit (lim⁡x→−2−f(x)=0 \lim_{x \to -2^-} f(x) = 0 ) and the right-hand limit (lim⁡x→−2+f(x)=−4 \lim_{x \to -2^+} f(x) = -4 ) are not equal, the limit as x x approaches −2-2 does not exist. Thus, lim⁡x→−2f(x)=DNE \lim_{x \to -2} f(x) = \text{DNE}

STEP 6

Identify the limit of f(x) f(x) as x x approaches −1-1.
From the graph, as x x approaches −1-1 from both the left and the right, f(x) f(x) approaches -4. The graph is continuous at x=−1 x = -1 . Thus, lim⁡x→−1f(x)=−4 \lim_{x \to -1} f(x) = -4
Solution: (a) lim⁡x→−2−f(x)=0 \lim_{x \to -2^-} f(x) = 0 \\ (b) lim⁡x→−2+f(x)=−4 \lim_{x \to -2^+} f(x) = -4 \\ (c) lim⁡x→−2f(x)=DNE \lim_{x \to -2} f(x) = \text{DNE} \\ (d) lim⁡x→−1f(x)=−4 \lim_{x \to -1} f(x) = -4

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