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h(x)={[()/(2x), for x <= -2],[2^(x), for -2 < x <= 0]:} Find lim_(x rarr-2)h(x) Choose 1 answer: (A) -2 (B) -(1)/(4) (C) (1)/(4)

\(h(x)=\left\{\begin{array}{ll}\frac{}{2 x} & \text { for } x \leq-2 \\2^{x} & \text { for }-2

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Q. h(x)={2x for x≤−22x for −2<x≤0h(x)=\left\{\begin{array}{ll}\frac{}{2 x} & \text { for } x \leq-2 \\2^{x} & \text { for }-2<x \leq 0\end{array}\right.\newlineFind lim⁡x→−2h(x) \lim _{x \rightarrow-2} h(x) \newlineChoose 11 answer:\newline(A) −2-2\newline(B) −14 -\frac{1}{4} \newline(C) 14 \frac{1}{4}
  1. Identify Function Definitions: Identify the function definitions around x=−2x = -2. h(x)=(12x)h(x) = \left(\frac{1}{2x}\right) for x≤−2x \leq -2 h(x)=2xh(x) = 2^x for -2 < x \leq 0
  2. Calculate Left-hand Limit: Calculate the left-hand limit as x x approaches −2 -2 .
    lim⁡x→−2−h(x)=lim⁡x→−2−(12x) \lim_{{x \to -2^-}} h(x) = \lim_{{x \to -2^-}} \left( \frac{1}{{2x}} \right)
    =12⋅(−2) = \frac{1}{{2 \cdot (-2)}}
    =1−4 = \frac{1}{{-4}}
    =−14 = -\frac{1}{{4}}
  3. Calculate Right-hand Limit: Calculate the right-hand limit as x x approaches −2 -2 .
    lim⁡x→−2+h(x)=lim⁡x→−2+2x \lim_{{x \to -2^+}} h(x) = \lim_{{x \to -2^+}} 2^x
    =2−2 = 2^{-2}
    =122 = \frac{1}{2^2}
    =14 = \frac{1}{4}
  4. Compare Limits: Compare the left-hand limit and right-hand limit. Left-hand limit: −14-\frac{1}{4} Right-hand limit: 14\frac{1}{4} Since −14≠14-\frac{1}{4} \neq \frac{1}{4}, the limit does not exist.

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