Math  /  Calculus

Question\begin{tabular}{|l|l|} \hlinef(3)=3f(3)=3 & limx3f(x)=2\lim _{x \rightarrow 3} f(x)=2 \\ \hlineg(3)=8g(3)=8 & limx3g(x)=8\lim _{x \rightarrow 3} g(x)=8 \\ \hlineh(3)=4h(3)=4 & limx3h(x)=2\lim _{x \rightarrow 3} h(x)=2 \\ \hline \end{tabular}
The table above gives selected values and limits of the functions f,gf, g, and hh. What is limx3(h(x)(2f(x)+3g(x)))?\lim _{x \rightarrow 3}(h(x)(2 f(x)+3 g(x))) ?

Studdy Solution

STEP 1

1. We are given the values and limits of the functions f(x)f(x), g(x)g(x), and h(x)h(x) at x=3x = 3.
2. The limit we need to find involves the product and sum of these functions as xx approaches 3.
3. Standard limit properties (e.g., limits of sums, products) can be used to simplify the calculation.

STEP 2

1. Apply the limit to the product h(x)(2f(x)+3g(x))h(x) \cdot (2f(x) + 3g(x)) as xx approaches 3.
2. Use the limit properties to separate the product and sum.
3. Substitute the known limits into the expression.
4. Simplify the resulting expression to find the final limit.

STEP 3

Write the given limit expression explicitly.
limx3(h(x)(2f(x)+3g(x))) \lim_{x \rightarrow 3} (h(x) \cdot (2f(x) + 3g(x)))

STEP 4

Apply the limit to the product using the limit property limxa[f(x)g(x)]=(limxaf(x))(limxag(x))\lim_{x \to a} [f(x) \cdot g(x)] = (\lim_{x \to a} f(x)) \cdot (\lim_{x \to a} g(x)).
limx3h(x)limx3(2f(x)+3g(x)) \lim_{x \rightarrow 3} h(x) \cdot \lim_{x \rightarrow 3} (2f(x) + 3g(x))

STEP 5

Apply the limit to the sum inside the product using the limit property limxa[f(x)+g(x)]=limxaf(x)+limxag(x)\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x).
limx3h(x)(limx32f(x)+limx33g(x)) \lim_{x \rightarrow 3} h(x) \cdot \left( \lim_{x \rightarrow 3} 2f(x) + \lim_{x \rightarrow 3} 3g(x) \right)

STEP 6

Substitute the known limits from the given information into the expression.
limx3h(x)(2limx3f(x)+3limx3g(x)) \lim_{x \rightarrow 3} h(x) \cdot \left( 2 \lim_{x \rightarrow 3} f(x) + 3 \lim_{x \rightarrow 3} g(x) \right) =22+38 = 2 \cdot 2 + 3 \cdot 8 =4+24 = 4 + 24 =28 = 28

STEP 7

Substitute the limit of h(x)h(x) as xx approaches 3.
limx3h(x)=2 \lim_{x \rightarrow 3} h(x) = 2 228 2 \cdot 28 =56 = 56
The solution to the limit limx3(h(x)(2f(x)+3g(x)))\lim_{x \rightarrow 3}(h(x)(2 f(x)+3 g(x))) is 5656.

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