l'Hôpital's rule with continuous function

May 2016
101
1
Ireland
Screenshot (5).png

hello
I am stuck on question 2
I know that the rule for a function to be continuous at a point it must be differentiable at that point.
I am not sure how I can use this to work out the problem
 

romsek

Math Team
Sep 2015
2,773
1,552
USA
A function continuous at a point need not be differentiable there. Consider $f(x) = |x|,~ @ x=0$

The limits of the continuous function at that point from both directions must agree however.

From the left it's pretty clear $f(0) = k$

Your job is to find $\lim \limits_{x \to 0+} \dfrac{e^{2x}-\cos(3x)}{2x^2+4x}$ which can be found using L'Hopital's rule.

I leave this to you.

Set $k$ to that limit from the right and Bob's yer uncle.
 
Last edited:
May 2016
101
1
Ireland
A function continuous at a point need not be differentiable there. Consider $f(x) = |x|,~ @ x=0$

The limits of the continuous function at that point from both directions must agree however.

From the left it's pretty clear $f(0) = k$

Your job is to find $\lim \limits_{x \to 0+} \dfrac{e^{2x}-\cos(3x)}{2x^2+4x}$ which can be found using L'Hopital's rule.

I leave this to you.

Set $k$ to that limit from the right and Bob's yer uncle.
thank you i get it now