Limit $\lim_{u\to0} \frac{3u}{\tan 2u}$

Question

I’m currently stuck trying to evaluate this limit, $$ \lim_{u\to0} \frac{3u}{\tan(2u)}, $$ without using L’Hôpital’s rule. I’ve tried both substituting for $\tan(2u)=\dfrac{2\tan u}{1-(\tan u)^2}$, and $\tan 2u=\dfrac{\sin 2u}{\cos 2u}$ without success. Am I on the right path to think trig sub?

Answer 1

The key point is the strandard limit as $x\to 0 \,\frac{\sin x}x\to 1$, indeed we have that

$$\dfrac{3u}{\tan(2u)}=\dfrac{3u}{2u}\dfrac{2u}{\tan(2u)}=\dfrac{3}{2}\dfrac{2u}{\sin(2u)}\cos (2u)\to \frac32\cdot 1 \cdot 1 = \frac32$$

with your first idea we obtain

$$\dfrac{3u}{\tan(2u)}=\dfrac{3u}{2\tan(u)}(1-(\tan u)^2))=\frac32\frac u {\sin u}\cos u(1-(\tan u)^2))\to \frac32\cdot 1\cdot1\cdot 1=\frac32$$

Refer to the related

• How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

Answer 2

$$ \begin{aligned} \lim_{u\to 0}\frac{3u}{\tan(2u)}&=\lim_{u\to 0}\frac{3u\cos(2u)}{\sin(2u)}\\ &=\lim_{u\to 0}\frac{3\cos(2u)}{2}\cdot\frac{2u}{\sin(2u)}\\ &=\left(\lim_{u\to 0}\frac{3\cos(2u)}{2}\right)\cdot \left(\lim_{u\to 0}\frac{2u}{\sin(2u)}\right)\qquad\text{since both limits exist}\\ &=\frac{3}{2}\cdot 1\\ &=\frac{3}{2}. \end{aligned} $$

Answer 3

Note that $$ \lim_{u\to 0}\frac{3u}{\tan(2u)}=\frac{3}{2}\times\lim_{u\to 0}\frac{2u}{\sin(2u)}\times\lim_{u\to 0}\cos(2u). $$ Now use your knowledge of well-known limits.

Answer 4

Just Taylor the $\tan(2u)$ and the answer comes straight away after you divide by $u$ both the numerator and denominator. This is assuming that $u$ tends to $0$. With infinity limit does not exist.

Answer 5

Because of Taylor series, $\tan2u\sim2u, \quad u\to0$. Then

$$\lim_{u\to0}\frac{3u}{2u}=\frac32$$

Answer 6

Hint:

$$\lim_{u\to0} \frac{3u}{\tan(2u)} = \frac{3}{2}\cdot\lim_{u\to0} \bigg[\frac{2u}{\sin(2u)}\cdot\cos (2u)\bigg]$$

Recall that $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

and apply it here.

Answer 7

By MVT

$$\tan(2u)-\tan(0)=$$ $$(2u-0)\Bigl(1+\tan^2(c)\Bigr)=$$

$$2u(1+\tan^2(c))$$

when $u\to 0, \; c\to 0$.

the limit is then $$\frac 32$$

Answer 8

$$\lim_{x\to 0} {3x\over \tan 2x}=\lim_{x\to 0} {3x\cos 2x\over \sin 2x}=\lim_{x\to 0} {3x\over \sin 2x}={3\over 2}$$