Question

question: 4
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use the following information to evaluate the given limit, when possible.
\\(\lim_{x\to9}f(x)=6\\) \\(\lim_{x\to6}f(x)=9\\) \\(f(9)=6\\)
\\(\lim_{x\to9}g(x)=3\\) \\(\lim_{x\to6}g(x)=3\\) \\(g(6)=3\\)
\\(\lim_{x\to9}g(f(x))\\)
if you need to enter \\(\infty\\), you may type infinity, or just inf. if the limit does not exist, you may type does not exist, or just dne. if it is may type not possible to know, or just npk.

Explanation:

Step1: Recall limit - composition rule

If $\lim_{x
ightarrow a}f(x)=L$ and $g(x)$ is continuous at $x = L$, then $\lim_{x
ightarrow a}g(f(x))=g(\lim_{x
ightarrow a}f(x))$.
We are given $\lim_{x
ightarrow9}f(x)=6$ and $\lim_{x
ightarrow6}g(x)=3$.

Step2: Apply the limit - composition rule

Since $\lim_{x
ightarrow9}f(x)=6$ and $\lim_{x
ightarrow6}g(x)$ exists, we can find $\lim_{x
ightarrow9}g(f(x))$ by substituting the limit of $f(x)$ into $g(x)$.
So, $\lim_{x
ightarrow9}g(f(x))=\lim_{y
ightarrow6}g(y)$ (where $y = f(x)$ and as $x
ightarrow9$, $y = f(x)
ightarrow6$).
Since $\lim_{x
ightarrow6}g(x)=3$, we have $\lim_{x
ightarrow9}g(f(x)) = 3$.

Answer:

3