QuestionFind the limit: $\lim _{x \rightarrow 2}[f(x)+5 g(x)]$ given $\lim _{x \rightarrow 2} f(x)=9$ and $\lim _{x \rightarrow 2} g(x)=-5$ .

Studdy Solution

STEP 1

Assumptions1. The limit of $f(x)$ as $x$ approaches is9. . The limit of $g(x)$ as $x$ approaches is -5.
3. The limit of $h(x)$ as $x$ approaches is0.
4. We are asked to find the limit of the expression $[f(x)+5 g(x)]$ as $x$ approaches.

STEP 2

We can use the limit laws to solve this problem. According to the limit laws, the limit of a sum or difference of two functions is the sum or difference of their limits. So, we can write the limit of the expression $[f(x)+5 g(x)]$ as the sum of the limits of $f(x)$ and $5g(x)$ .
$\lim{x \rightarrow2}[f(x)+5 g(x)] = \lim{x \rightarrow2}f(x) +5\lim{x \rightarrow2}g(x)$

STEP 3

Now, substitute the given limits of $f(x)$ and $g(x)$ into the equation.
$\lim{x \rightarrow2}[f(x)+5 g(x)] =9 +5(-5)$

STEP 4

Perform the calculation to find the limit.
$\lim{x \rightarrow2}[f(x)+ g(x)] =9 -25 = -16$ So, the limit of the expression $[f(x)+ g(x)]$ as $x$ approaches2 is -16.

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