1. ## Trigonometric Limits

Hi all, I am having trouble simplifying or factoring the following trigonometric limits. I can't find the identities for the last two.

1. lim (1-tan(x))/(sin(x)-cos(x)) as x approaches pie/4

2. lim (sin(3t))/(2t) as t approaches 0

3. lim (sin(2x))/(sin(3x)) as x approaches 0

Any help would be appreciated!

2. Originally Posted by Enderless
Hi all, I am having trouble simplifying or factoring the following trigonometric limits. I can't find the identities for the last two.

1. lim (1-tan(x))/(sin(x)-cos(x)) as x approaches pie/4

2. lim (sin(3t))/(2t) as t approaches 0

3. lim (sin(2x))/(sin(3x)) as x approaches 0

Any help would be appreciated!
what have you tried so far?

for the first, change everything to be in terms of sine and cosine and see what cancels out

for the second, use the special result. $\displaystyle \lim_{x \to 0} \frac {\sin x}{x} = 1$

for the third, expand the sines with the double angle formula and see what cancels out.

3. Originally Posted by Enderless
Hi all, I am having trouble simplifying or factoring the following trigonometric limits. I can't find the identities for the last two.

1. lim (1-tan(x))/(sin(x)-cos(x)) as x approaches pie/4

2. lim (sin(3t))/(2t) as t approaches 0

3. lim (sin(2x))/(sin(3x)) as x approaches 0

Any help would be appreciated!
2. and 3. are trivial just expand the sin's as power series (you will just need the first non zero term).

RonL

4. For 2 and 3 use this:
$\displaystyle \displaystyle\lim_{x\to 0}\frac{\sin\alpha x}{\beta x}=\frac{\alpha}{\beta}$ and $\displaystyle \lim_{x\to 0}\frac{\sin\alpha x}{\sin\beta x}=\frac{\alpha}{\beta}$.

Proof:
$\displaystyle \lim_{x\to 0}\frac{\sin\alpha x}{\beta x}=\lim_{x\to 0}\frac{\sin\alpha x}{\alpha x}\cdot\frac{\alpha}{\beta}=\frac{\alpha}{\beta}$.

$\displaystyle \lim_{x\to 0}\frac{\sin\alpha x}{\sin\beta x}=\lim_{x\to 0}\frac{\sin\alpha x}{\alpha x}\cdot\frac{\beta x}{\sin\beta x}\cdot\frac{\alpha}{\beta}=\frac{\alpha}{\beta}$.

5. Going over my work, I've found the solutions to the problems already before Jhevon posted a reply. It turns out that it was easier than I thought by using the triple angle identities or something like that.

Thanks for the replies everyone!

6. Originally Posted by Enderless
Going over my work, I've found the solutions to the problems already before Jhevon posted a reply. It turns out that it was easier than I thought by using the triple angle identities or something like that.

Thanks for the replies everyone!
Good job! and good luck with your class