Tuesday

September 2, 2014

September 2, 2014

Posted by **mike** on Monday, December 6, 2010 at 7:59pm.

lim (cos(x))^(7/x^2) as x goes x-->0^+

- L'hospital rule -
**MathMate**, Monday, December 6, 2010 at 8:16pmL'Hôpital's rule applies when there is a fraction whose numerator and denominator are both undefined or zero.

Since the given expression is not a fraction, we need to transform it to a form where L'Hôpital's rule applies.

Taking log is a good way when powers are involved:

ln((cos(x))^(7/x^2))

=(7/x²)*ln(cos(x))

=7ln(cos(x))/x²

Now both numerator and denominator become zero as x->x+.

differentiate with respect to x:

7(-sin(x)/cos(x)) / 2x

=-7tan(x)/2x

Since the expression is still undefined when x->0+, we can differentiate again:

-7sec²(x)/2

=-7/2 as x->0+

So the original limit can be found by raising -7/2 to the power of e, or

Lim x->0+ cos(x)^(7/x²)

= e^{-7/2}

**Answer this Question**

**Related Questions**

Calculus - Evaluate the limit using L'Hospital's rule if necessary. lim as x ...

math - i need some serious help with limits in pre-calc. here are a few ...

Calculus - L'Hospital's Rule Lim(lnx)^(x-1) x->1+ This is what I did so far...

Calculus Help Please Urgent!!! - find the limit algebraically. Use L'Hospital's ...

math - how is the limit of x^2/(1-cos^2(x)) as x goes to 0 is 1. I know how to ...

Evaluating limits - Evaluate the limit. (If you need to use - or , enter -...

calculus - using L'Hospital's rule, evalutate; LIM as x->0 e^x +cos x / e^x...

Calculus 1 - Evaluate the following limits after identifying the indeterminate ...

calculus - Evaluate the following limits after identifying the indeterminate ...

Calculus, please check my answers! - 1. Evaluate: lim x->infinity(x^4-7x+9)/(...