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)]

Solution Steps

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

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

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