Questions: [ lim x rightarrow 2 mathrmf(mathrmx)=4 quad lim mathrmx rightarrow 2 mathrm~g(mathrmx)=-2 quad lim mathrmx rightarrow 2 mathrm~h(mathrmx) ] find the limits that exist. If the limit does not exist, explain why. a. quad lim x rightarrow 2[f(x)+5 g(x)]

[ lim x rightarrow 2 mathrmf(mathrmx)=4 quad lim mathrmx rightarrow 2 mathrm~g(mathrmx)=-2 quad lim mathrmx rightarrow 2 mathrm~h(mathrmx) ] find the limits that exist. If the limit does not exist, explain why. a. quad lim x rightarrow 2[f(x)+5 g(x)]
Transcript text: \[ \lim _{x \rightarrow 2} \mathrm{f}(\mathrm{x})=4 \quad \lim _{\mathrm{x} \rightarrow 2} \mathrm{~g}(\mathrm{x})=-2 \quad \lim _{\mathrm{x} \rightarrow 2} \mathrm{~h}(\mathrm{x}) \] find the limits that exist. If the limit does not exist, explain why. a. $\quad \lim _{x \rightarrow 2}[f(x)+5 g(x)]$
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Solution

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Solution Steps

Step 1: Identify Given Limits

We are given the limits: \[ \lim_{x \rightarrow 2} f(x) = 4 \quad \text{and} \quad \lim_{x \rightarrow 2} g(x) = -2. \]

Step 2: Apply Limit Properties

Using the properties of limits, we can express the limit of the sum as: \[ \lim_{x \rightarrow 2} [f(x) + 5g(x)] = \lim_{x \rightarrow 2} f(x) + \lim_{x \rightarrow 2} [5g(x)]. \]

Step 3: Calculate Each Limit

Substituting the known limits into the expression, we have: \[ \lim_{x \rightarrow 2} [5g(x)] = 5 \cdot \lim_{x \rightarrow 2} g(x) = 5 \cdot (-2) = -10. \] Thus, the overall limit becomes: \[ \lim_{x \rightarrow 2} [f(x) + 5g(x)] = 4 + (-10) = -6. \]

Final Answer

\(\boxed{-6}\)

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