# Given $\lim \limits_{x \to c}f(g(x)) = f(\lim \limits_{x \to c}g(x))$ provided what 2 conditions are met?

The problem that I am having trouble with is $$\lim \limits_{x \to c}f(g(x)) = f(\lim \limits_{x \to c}g(x))$$. It seems like the limit of function $$g$$ is wrapped inside of function $$f$$. I need to provide two conditions that make this statement true. I am guessing that I have to use two of the three rules from "How to determine whether or not a function is continuous" but I am not sure. However, I know that $$f(c)$$ is not defined.

• You will need continuity of $f$ at $L=\lim_{x\to c} g(x)$. Mar 16 at 9:00

The conditions are: $$f\circ g$$ and $$g$$ are both continuous at $$x = c$$. To see this is the case, we need to have: $$\displaystyle \lim_{x\to c} f(g(x)) = \displaystyle \lim_{x \to c} (f\circ g)(x) = (f\circ g)(c) = f(g(c)) = f\left(\displaystyle \lim_{x \to c} g(x) \right)$$. So to make it happen, you need what is stated above.

• It is not required that $g$ be continuous or even defined at $x = c$. It is only required that $\lim_{x \rightarrow c} g(x)$ exist. Mar 16 at 7:15
• Great. Is there any resource for me to grasp this because even I know the answer now. However, I still do not know what the given means or how to even come up with those conditions. Mar 16 at 7:29
• @MichaelGuest: I just explained it in the edited post. Please see it,and let me know if you find it right or not...
– user899577
Mar 16 at 7:55
• Great thank you very much. I understand it better now. Mar 16 at 9:07

Let $$A,B, C$$ be non empty subsets of $$\mathbb R$$ and $$g:A\to B, f:g(A)\to C$$, be two functions, where $$g(A)=\{g(x):x\in A\}$$. Clearly, $$g(A)\subseteq B$$.

With this condition $$fog:A\to C$$ is defined.

Now let $$c\in \mathbb R$$ be a limit point of $$A$$. If

1. $$\lim_{x\to c}g(x)$$ exists.

and

2. $$f$$ is continuous at $$\lim_{x\to c}g(x)$$.

Then, $$\lim_{x\to c}f(g(x))=f(\lim_{x\to c}g(x))$$

• Great thank you. This helps a lot. Mar 16 at 16:07