- #1

- 274

- 1

I recently learned about L'Hopital's Rule in calculus and as I was doing some practice problems I came across two that confused me a lot. I was hoping that someone could help me with them here.

The problem is lim

I tried to make (sint)(lnt) a quotient by setting the problem up as: lim

This made it so that both the top and the bottom of the function undefined when I plug in zero.

I thought maybe I would be able to differentiate both the top and the bottom and see if maybe that would lead me somewhere but it only made it worse by coming out as lim

I always thought that because sin(0) is equal to 0 and since 0* anything is equal to 0 even if ln(0) is undefined then the limit of this equation is equal to 0. I don’t really trust that reasoning even though it sounds reasonable because it came out of my own head haha, and I’m not sure how I would show that as my work on a test so could you please give me some tips on how to proceed?

I also had a problem with lim

The way I did it was I took the function x^(1/x) and set it = y. I then took the ln of both sides so that I could make it lny = 1/xlnx

I then plugged that back into the limit to get:

lim

I then used e to find the limit by:

lim

If this isn’t the right way to do the problem, could you please explain to me how to do it?

Thanks,

Aaron Wong

WOW. I just looked over my post and it came out all funky. I'll try to clear things up if anything isn't understandable but if someone could point me in the right direction on how to post equations on the forum I'd be very grateful. Thanks.

The problem is lim

_{(t→0)}〖(sint)(lnt)〗.I tried to make (sint)(lnt) a quotient by setting the problem up as: lim

_{(t→0)}〖(lnt)/(csct)〗This made it so that both the top and the bottom of the function undefined when I plug in zero.

I thought maybe I would be able to differentiate both the top and the bottom and see if maybe that would lead me somewhere but it only made it worse by coming out as lim

_{(t→0)}〖x^(-1)/(-csctcott)〗I always thought that because sin(0) is equal to 0 and since 0* anything is equal to 0 even if ln(0) is undefined then the limit of this equation is equal to 0. I don’t really trust that reasoning even though it sounds reasonable because it came out of my own head haha, and I’m not sure how I would show that as my work on a test so could you please give me some tips on how to proceed?

I also had a problem with lim

_{(x→∞)}〖x^(1/x) 〗The way I did it was I took the function x^(1/x) and set it = y. I then took the ln of both sides so that I could make it lny = 1/xlnx

I then plugged that back into the limit to get:

lim

_{(x→∞)}〖lny= lim_{(x→∞)}〖ln(x^(1/x) )= lim_{(x→∞)}〖1/x ln〖x= 1/∞〗 〗 〗 〗 ln∞=0∞=0I then used e to find the limit by:

lim

_{(x→∞)}〖x^(1/x)= lim_{(x→∞)}〖e^(ln(x^(1/x)))= e^lim_{(x→∞)}〖ln(x^(1/x))〗 = e^0=1〗 〗If this isn’t the right way to do the problem, could you please explain to me how to do it?

Thanks,

Aaron Wong

WOW. I just looked over my post and it came out all funky. I'll try to clear things up if anything isn't understandable but if someone could point me in the right direction on how to post equations on the forum I'd be very grateful. Thanks.

Last edited: