Determining limits with use of L'Hopital's rule

Mar 2013
56
0
Hallo!

Please help, i am struggling with these questions for revisions, and your solutions would be invaluable to me.

Need to find the answers to these questions WITHOUT L'Hopital's rule

A) lim x-> 0 (x^2 + |x|) cos (pi/x)

B) lim x->0 ((e^(2x) - 1)/(e^(3x) - 1))



thank you in advance!
 

greg1313

Forum Staff
Oct 2008
8,008
1,174
London, Ontario, Canada - The Forest City
B) Make the substitution \(\displaystyle u\,=\,e^{3x}\,-\,1\) so the limit becomes \(\displaystyle \lim_{u\to0}\,\frac{(u\,+\,1)^{\frac23}\,-\,1}{u}\)
Applying the binomial series to \(\displaystyle (u\,+\,1)^{\frac23}\) leads to a limit of 2/3 after some elementary manipulations.

Is A) typed correctly?
 
Mar 2013
56
0
Oh my gosh, im sorry. i typed it wrong

A) lim x-> 0 (x^2 + |x|) cos (pi/x)

thank you!
 
Feb 2013
281
0
Question A) is still suspicious. You alluded that the question can be solved by applying l'Hospital's rule, that is "0/0" or "inf/inf" type. But it is actually not. It's "0*small_number" type.
 
Mar 2013
56
0
Im sorry if it is unclear but that is all the info i was given
 

zaidalyafey

Math Team
Aug 2012
1,177
44
Sana'a , Yemen
razzatazz said:
Oh my gosh, im sorry. i typed it wrong

A) lim x-> 0 (x^2 + |x|) cos (pi/x)

thank you!
You can solve it by squeeze theorem ...
 

zaidalyafey

Math Team
Aug 2012
1,177
44
Sana'a , Yemen
The topic and the question contradicts each other !
 

HallsofIvy

Math Team
Sep 2007
2,409
6
razzatazz said:
Hallo!

Please help, i am struggling with these questions for revisions, and your solutions would be invaluable to me.

Need to find the answers to these questions WITHOUT L'Hopital's rule

A) lim x-> 0 (x^2 + |x|) cos (pi/x)
You are aware that cosine is always between -1 and 1, are you not?

B) lim x->0 ((e^(2x) - 1)/(e^(3x) - 1))
Divide both numerator and denominator by \(\displaystyle e^{3x}\).

thank you in advance!
 

HallsofIvy

Math Team
Sep 2007
2,409
6
zaidalyafey said:
The topic and the question contradicts each other !
Yes, he clearly meant to title this "without the use of L'Hopital".