h(x)={[5x," for "x = -2]:} Find lim_(x rarr-2)h(x). Choose 1 answer: (A) -10 (B) -2 (C) 10 (D) The limit doesn't exist.

Consider Limits: To find the limit of the piecewise function h(x) as x approaches -2, we need to consider the limit from both sides of -2, that is, from the left and from the right.Left Side Limit: First, let's find the limit from the left side (x approaching -2 from values less than -2). For x -2^-) h(x) = lim_(x -> -2^-) 5x = 5 * (-2) = -10.Right Side Limit: Now, let's find the limit from the right side (x approaching -2 from values greater than or equal to -2). For x >= -2, the function is defined as h(x) = x^3 - 2. lim_(x -> -2^+) h(x) = lim_(x -> -2^+) (x^3 - 2) = (-2)^3 - 2 = -8 - 2 = -10.Final Limit: Since the limit from the left side and the limit from the right side are equal, the limit of h(x) as x approaches -2 exists and is equal to -10.

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Math Problems

Calculus

Find derivatives of logarithmic functions

Full solution

h(x)=\left\{\begin{array}{ll} 5 x & \text { for } x<-2 \\ x^{3}-2 & \text { for } x \geq-2 \end{array}\right.

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Find

\lim _{x \rightarrow-2} h(x)

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Choose

1

answer:

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(A)

-10

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(B)

-2

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(C)

10

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(D) The limit doesn't exist.

Consider Limits: To find the limit of the piecewise function $h(x)$ as $x$ approaches $-2$ , we need to consider the limit from both sides of $-2$ , that is, from the left and from the right.
Left Side Limit: First, let's find the limit from the left side $x$ approaching $-2$ from values less than $-2$ . For x < -2, the function is defined as $h(x) = 5x$ . $\lim_{x \to -2^-} h(x) = \lim_{x \to -2^-} 5x = 5 \times (-2) = -10$ .
Right Side Limit: Now, let's find the limit from the right side $x$ approaching $-2$ from values greater than or equal to $-2$ . For $x \geq -2$ , the function is defined as $h(x) = x^3 - 2$ . $\lim_{x \to -2^+} h(x) = \lim_{x \to -2^+} (x^3 - 2) = (-2)^3 - 2 = -8 - 2 = -10$ .
Final Limit: Since the limit from the left side and the limit from the right side are equal, the limit of $h(x)$ as $x$ approaches $-2$ exists and is equal to $-10$ .