L'Hôpital's Rule

[eggmath]

Question:
If f'(x)=cos(x) and g′(x) = 1 for all x, and if f(0)=g(0)=0,
lim x→∞ (f(x)/g(x)) is ?

When I try solving using l'hôpital's rule, I get lim x→∞ cos (x), but then that would mean that the limit is nonexistent.
Did I do something wrong? or does the limit actually not exist?

Also, I'm confused why the value for f(0) and g(0) is relevant when the question is asking for the limit as x approaches infinity and not 0.
Would you need to first find f(x) and g(x) by integrating the given f'(x) and g'(x)? Is that why they give you f(0) and g(0)?

Thanks!

 

[Harry_the_cat]

Yes you need to find f(x) and g(x).

f(0)=g(0)=0 helps you find "c".

L’Hospital’s Rule applies if we have an indeterminate form 0/0 or ∞/∞. It doesn't apply here.

 

[eggmath]

Ok, so if you integrate f(x) and g(x), you get that the original limit is:
lim x→∞ (sin(x)/x)

which would be 0, correct?

 

[Harry_the_cat]

Yes.

 

[Steven G]

Yes you need to find f(x) and g(x).

f(0)=g(0)=0 helps you find "c".

L’Hospital’s Rule applies if we have an indeterminate form 0/0 or ∞/∞. It doesn't apply here.

To eggmath: While it is true that f(0) = g(0) = 0 will help you find the two integration constants they are not needed to evaluate the limit.