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QuestionFind lim⁡x→2(h(x)(5f(x)+g(x)))\lim _{x \rightarrow 2}(h(x)(5 f(x)+g(x))) given f(2)=3f(2)=3, g(2)=−6g(2)=-6, h(2)=−3h(2)=-3, limits at x=2x=2.

Studdy Solution

STEP 1

Assumptions1. The function values and limits at x=x= for f(x)f(x), g(x)g(x), and h(x)h(x) are given in the table. . We need to find the limit as xx approaches for the function h(x)(5f(x)+g(x))h(x)(5f(x)+g(x)).

STEP 2

We know that the limit of a product of two functions is the product of their limits, and the limit of a sum of two functions is the sum of their limits. Therefore, we can rewrite the limit we need to find as followslim⁡x→2(h(x)(5f(x)+g(x)))=lim⁡x→2h(x)⋅lim⁡x→2(5f(x)+g(x))\lim{x \rightarrow2}(h(x)(5 f(x)+g(x))) = \lim{x \rightarrow2}h(x) \cdot \lim{x \rightarrow2}(5 f(x)+g(x))

STEP 3

We can further break down the second limit as followslim⁡x→2(5f(x)+g(x))=5⋅lim⁡x→2f(x)+lim⁡x→2g(x)\lim{x \rightarrow2}(5 f(x)+g(x)) =5 \cdot \lim{x \rightarrow2}f(x) + \lim{x \rightarrow2}g(x)

STEP 4

Now we can substitute the given limits and function values into the equation.lim⁡x→2h(x)=2\lim{x \rightarrow2}h(x) =2lim⁡x→2f(x)=4\lim{x \rightarrow2}f(x) =4lim⁡x→2g(x)=−6\lim{x \rightarrow2}g(x) = -6So,
lim⁡x→2(h(x)(f(x)+g(x)))=2⋅(⋅4−6)\lim{x \rightarrow2}(h(x)( f(x)+g(x))) =2 \cdot ( \cdot4 -6)

STEP 5

Now we can calculate the value of the limit.
lim⁡x→2(h(x)(5f(x)+g(x)))=2⋅(20−)=2⋅14=28\lim{x \rightarrow2}(h(x)(5 f(x)+g(x))) =2 \cdot (20 -) =2 \cdot14 =28So, the limit of the function h(x)(5f(x)+g(x))h(x)(5f(x)+g(x)) as xx approaches2 is28.
The answer is (C)28.

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