# Thread: limits of trigonometric fxns...?

1. ## limits of trigonometric fxns...?

okay so i'm confused as to HOW to go about finding the limits of these trigonometric fxns....

lim x-->0 tan(2x) / sin(3x)cos(4x)

and

lim x-->0 sin(x^2 - 9) / x-3

2. Originally Posted by veezey11
okay so i'm confused as to HOW to go about finding the limits of these trigonometric fxns....

lim x-->0 tan(2x) / sin(3x)cos(4x)
We should use the fact that $\displaystyle \lim_{u\to0}\frac{\tan u}{u}=1$ and $\displaystyle \lim_{u\to0}\frac{\sin u}{u}=1$

Thus, $\displaystyle \lim_{x\to 0}\frac{\tan(2x)}{\sin(3x)\cos(4x)}=\lim_{x\to0}\f rac{\tan(2x)}{\sin(3x)}\cdot\underbrace{\lim_{x\to 0}\frac{1}{\cos(4x)}}_1=\lim_{x\to0}\frac{\display style\frac{\tan(2x)}{x}}{\displaystyle\frac{\sin(3 x)}{x}}=\lim_{x\to0}\frac{\displaystyle\frac{\tan( 2x)}{x}\cdot{\color{red}\frac{2}{2}}}{\displaystyl e\frac{\sin(3x)}{x}\cdot{\color{red}\frac{3}{3}}}$ $\displaystyle =\lim_{x\to0}\frac{2\displaystyle\frac{\tan(2x)}{2 x}}{3\displaystyle\frac{\sin(3x)}{3x}}=\frac{2}{3} \frac{\displaystyle\lim_{x\to0}\frac{\tan(2x)}{2x} }{\displaystyle\lim_{x\to0}\frac{\sin(3x)}{3x}}=\f rac{2}{3}\cdot\frac{1}{1}=\color{red}\boxed{\frac{ 2}{3}}$

Does this make sense?

--Chris

3. uhhhm sortof yeah... i'll take a closer look at it. and see if i can figure it all out. THANK YOU SO MUCH!

<3

4. Originally Posted by veezey11
okay so i'm confused as to HOW to go about finding the limits of these trigonometric fxns....

lim x-->0 tan(2x) / sin(3x)cos(4x)

and

lim x-->0 sin(x^2 - 9) / x-3
The easier way is with L'Hospital's rule, namely

$\displaystyle \lim_{x \to a} \frac{g(x)}{h(x)} = \lim_{x \to a} \frac{g'(x)}{h'(x)}$.

So $\displaystyle \lim_{x \to 0} \frac{\tan{2x}}{\sin{3x}\cos{4x}} = \lim_{x \to 0} \frac{\frac{d}{dx}(\tan{2x})}{\frac{d}{dx}(\sin{3x }\cos{4x})}$

$\displaystyle = \lim_{x \to 0} \frac{\frac{2}{\cos^2{2x}}}{3\cos{3x}\cos{4x} - 4\sin{3x}\sin{4x}}$

$\displaystyle = \lim_{x \to 0} \frac{2}{\cos^2 {2x}[3\cos{3x}\cos{4x} - 4\sin{3x}\sin{4x}]}$

Can you go from here?

5. Originally Posted by veezey11
okay so i'm confused as to HOW to go about finding the limits of these trigonometric fxns....

lim x-->0 tan(2x) / sin(3x)cos(4x)

and

lim x-->0 sin(x^2 - 9) / x-3
For the second, why can't you just use direct substitution?