Evaluating Limits!!!!!

I'm having a few problems with calculating a few (what I call) tricky limits. My teacher hasn't really explained the various methods of how to solve questions such as the one's below and I was wondering whether anybody could help me?

a) limit x--> 0 of (sin(4*x^(2)))/(x^(2)). The questions is to do with continuity and I must find a constant (c) for the function to be continuous at x = 0 therefore I must evaluate at x = 0

b) limit x--> 0 of (1-e^(-2x))/(sin(4x)). Again the question is asking the same as above. The questions is to do with continuity and I must find a constant (c) for the function to be continuous at x = 0 therefore I must evaluate at x = 0

Help and guidance would be much appreciated :)

so when you try to compute these limits you should get 0/0, which should be a red flag

fortunately, we have L'Hopital's rule: http://en.wikipedia.org/wiki/L%27H%C3%B4…

which states that we can take the derivative of the numerator and the derivative of the denominator and still get the same limit. sooo...

lim x->0 of (sin(4*x^(2)))/(x^(2))

= lim x->0 of (8x*cos(4x^2))/2x

= lim x->0 of ((-64x^2)*sin(4x^2) + 8*cos(4x^2)) / 2

= 8/2

= 4


The next one follows the same process, but here you only have to take the derivative once and then you can evaluate the limit:

limit x--> 0 of (1-e^(-2x))/(sin(4x))

= lim x->0 of (2e^(-2x))/(4*cos(4x))

= 2/4

= 1/2

Apply L'Hopitals Rule to each to get

a) 4

b) 1/2

For a discussion of L'Hopitals Rule, follow the link

http://en.wikipedia.org/wiki/L'H%C3%B4pi…