# Limit $\lim_{x\to 0^{+}} (\tan x)^x$

$$\lim_{x\to 0^{+}} (\tan x)^x$$

$$\lim_{x\to 0^{+}} e^{\ln((\tan x)^x)}=\lim_{x\to 0^{+}} e^{x\ln(\tan x)}=\lim_{x\to 0^{+}} e^{x[\ln(\sin x)-\ln(\cos x)]}$$

We can continue to create an expression that may help us use L'Hospital but it does not seem to be correct

P.S can we write:

$$1=(\frac{-1}{-1})^x\leq \lim_{x\to 0^+}\Bigl(\frac{\sin x}{\cos x}\Bigr)^x\leq \Bigl(\frac{1}{1}\Bigr)^x=1$$?

• You mean $(\tan x)^x$ or $\tan(x^x)$? Commented Feb 7, 2020 at 8:11
• sorry $(tanx)^x$ Commented Feb 7, 2020 at 8:15
• it would be better to write $$\left(1\right)^{x}=\left(\frac{-1}{-1}\right)^{x}\le\frac{\sin\left(x\right)^{x}}{\cos\left(x\right)^{x}}=\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)^{x}\le\left(1\right)^{x}$$ because $-1^{x}$ is not defined for all real $x$
– user715522
Commented Feb 7, 2020 at 8:22
• @user715522 I removed its as $(-1)^x\leq sin(x)^x$ not for all $x$ Commented Feb 7, 2020 at 8:23

$$\lim_{x \to 0+} x \ln (\tan x)=\lim_{x \to 0+} \frac {\ln (\tan x)} {1/x}=-\lim_{x \to 0+} \frac {\sec^{2}x} {\tan x /x^{2}}$$. You can write this as $$-\lim_{x \to 0+} \frac {x^{2}} { \sin x \cos x}$$ and this limit is $$0$$. (Why?). Hence the given limit is $$e^{0}=1$$.

$$\lim_{x\to0^+}(\tan x)^x=\left(\lim_{x\to0^+}\tan x^{\tan x}\right)^{\lim_{x\to^+}\dfrac x{\tan x}}$$

For the inner limit use Limit of $x^x$ as $x$ tends to $0$

Compute first the limit of the logarithm, using equivalence:

$$\tan x\sim_0 x,\:$$ so $$\quad \ln\bigl((\tan x)^x\bigr)=x\ln(\tan x)\sim_0x\ln x \xrightarrow[x\to 0]{} 0.$$

• You enjoy equivalents as much as I enjoy Taylor (which is the same). Cheers :-) Commented Feb 7, 2020 at 9:52
• I love very short proofs, without irrelevant details… ;-) Commented Feb 7, 2020 at 9:55

You surely know

• $$\lim_{x\to 0}\frac{\sin x}{x} = 1 \Rightarrow \lim_{x\to 0}\frac{\tan x}{x} = \lim_{x\to 0}\left(\frac{\sin x}{x}\cdot \frac 1{\cos x} \right) = 1$$.
• Besides this, it is easy to show that $$\lim_{x\to 0^+}x^x = 1$$.

Hence,

$$(\tan x)^x = \underbrace{\left(\frac{\tan x}{x}\right)^x}_{\stackrel{x\to 0^+}{\longrightarrow}1^0=1}\cdot \underbrace{x^x}_{\stackrel{x\to 0^+}{\longrightarrow}1} \stackrel{x\to 0^+}{\longrightarrow}1$$