Question

use the following information to evaluate the given limit, when possible.
$lim_{x
ightarrow9}f(x)=6$ $lim_{x
ightarrow6}f(x)=9$ $f(9)=6$
$lim_{x
ightarrow9}g(x)=3$ $lim_{x
ightarrow6}g(x)=3$ $g(6)=3$
$lim_{x
ightarrow6}f(g(x))$
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.

Explanation:

Step1: Recall limit - composition rule

If $\lim_{x
ightarrow a}g(x)=L$ and $f(x)$ is continuous at $x = L$, then $\lim_{x
ightarrow a}f(g(x))=f(\lim_{x
ightarrow a}g(x))$.
We want to find $\lim_{x
ightarrow 6}f(g(x))$. We know that $\lim_{x
ightarrow 6}g(x)=3$.

Step2: Evaluate $f$ at the limit of $g(x)$

We need to find $f(\lim_{x
ightarrow 6}g(x))$. Since $\lim_{x
ightarrow 6}g(x)=3$, and we don't have any information about the continuity of $f(x)$ at $x = 3$, but we can't directly apply the composition - rule. However, we note that we don't have enough information to suggest any non - continuity issues at the relevant points. So, we assume the conditions for the limit of the composition function are met. Then $\lim_{x
ightarrow 6}f(g(x))=f(\lim_{x
ightarrow 6}g(x))$. Substituting $\lim_{x
ightarrow 6}g(x)=3$ into $f$, we don't have the value of $f(3)$ directly given. But if we assume the standard rules of limits for composition hold (and since no information to the contrary is provided), we note that we made a wrong start.

Let's use the fact that for the limit of a composite function $\lim_{x
ightarrow a}f(g(x))$, we consider the behavior of $g(x)$ as $x
ightarrow a$ and then $f$ at the resulting value.
We know that $\lim_{x
ightarrow 6}g(x) = 3$. Now we need to find $f(3)$. Since we have no information about $f(3)$ and no way to relate the given limits $\lim_{x
ightarrow 9}f(x)=6,\lim_{x
ightarrow 6}f(x)=9,f(9) = 6,\lim_{x
ightarrow 9}g(x)=3,\lim_{x
ightarrow 6}g(x)=3,g(6)=3$ to $f(3)$.
The limit does not exist.

Answer:

does not exist