0
$\begingroup$

enter image description here

enter image description here

PROBLEM: Limits do not exist for both of the functions as $x \to -2$ but they say both functions still have fixed limit value equal to $5$. Is it one of the Limits Rules or more of common-sense approach ?

As you can see, for both $f(x), g(x)$ have different values for limits when approached from left and right sides, which mean they don't have a limit by definition. But if we add them together and take their limit it is $5$ for both left and right side. Hence, Khan Academy concludes this composite function does have a limit.

Can I assume that this is what it means:

Limits Rule for composite functions: if limits do not exist for 2 or more individual functions but added together they reach same value for both left-side and right-side limits, then limit exists.

$\endgroup$
  • $\begingroup$ Try drawing a graph of $g(x)+h(x)$ to see how it behaves. Recall that the limit of a function does not depend on how you describe it, but only on what the function value is at each point. $\endgroup$ – Henning Makholm Aug 13 '18 at 9:17
  • $\begingroup$ Thanks. Good idea to graph it, did not occur to me :) $\endgroup$ – Arnuld Aug 13 '18 at 10:19
1
$\begingroup$

Let $f(x)=g(x)+h(x)$, what we are interested would be the function $f$, rather than function $g$ or $h$.

We can then study $\lim_{x \to c^-}f(x)$ and $\lim_{x \to c^+}f(x)$ to decide if the limit exists at $c$. Limit need not exists at point $c$ for $g$ and $h$.

Remark:

The word composite usually refers to $f\circ g$, then that $f\circ g(x)=f(g(x))$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.