Thread: problem with limits

I had some math problems that was troublesome, and I would like to know what are the steps to those problems. Here are the problems: 1)lim x->0 (tan5x/tan2x) the ans. is 5/2. 2) lim x->0 (1-cos(x))/sin(x). The ans. is 0. 3) lim x->0 (sec x-1)/(x sec(x)) the ans. is 0

Originally Posted by driver327

I had some math problems that was troublesome, and I would like to know what are the steps to those problems. Here are the problems: 1)lim x->0 (tan5x/tan2x) the ans. is 5/2. 2) lim x->0 (1-cos(x))/sin(x). The ans. is 0. 3) lim x->0 (sec x-1)/(x sec(x)) the ans. is 0

Problem 1]


We can ignore the second factor because and does not change the value of the limit.

Thus, the problem reduces to find,


Note, we can write,


Now,

Similar reasoning says that,

Thus, we are left with 5/2.

Originally Posted by driver327

2) lim x->0 (1-cos(x))/sin(x). The ans. is 0.

Hint: Write

lim x->0 (sec x-1)/(x sec(x))

Hint: Multiply denominator and numerator by then apply the argument above.

I did mult. the cos(x) and it got me (1-cos(x)/x) = 0. Did I do this right? One more question, is it a matter of only getting sines and cosines in the equation like this? And thanks PerfectHacker 4 helping me out.

Originally Posted by driver327

I did mult. the cos(x) and it got me (1-cos(x)/x) = 0. Did I do this right?

Yes

One more question, is it a matter of only getting sines and cosines in the equation like this? And thanks PerfectHacker 4 helping me out.

Usually.