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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.

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

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    $\begingroup$ 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. $\endgroup$ Mar 16 at 7:15
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    $\begingroup$ 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. $\endgroup$ Mar 16 at 7:29
  • $\begingroup$ @MichaelGuest: I just explained it in the edited post. Please see it,and let me know if you find it right or not... $\endgroup$
    – user899577
    Mar 16 at 7:55
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    $\begingroup$ Great thank you very much. I understand it better now. $\endgroup$ Mar 16 at 9:07
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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, $\lim_{x\to c}f(g(x))=f(\lim_{x\to c}g(x))$

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  • $\begingroup$ Great thank you. This helps a lot. $\endgroup$ Mar 16 at 16:07

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