Animated Solution for Mathematics - Limits, Continuity and Differentiability: Find ${\lim }_{x\to 0}\{\tan (\pi /4+x){\}}^{1/x}.$

We are evaluating the limit:

When $x\to 0$, the base $\tan (\frac{\pi }{4}+x)$ approaches $\tan (\frac{\pi }{4})=1$. Simultaneously, the exponent $\frac{1}{x}$ approaches $\infty$.

This results in the indeterminate form $1^{\infty }$. To resolve this, we utilize the standard exponential transformation rule:

We focus on the term $f(x)-1$, where $f(x)=\tan (\frac{\pi }{4}+x)$. Applying the trigonometric identity $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$, we substitute $A=\frac{\pi }{4}$ and $B=x$:

Now, we perform the subtraction $f(x)-1$:

Simplifying the numerator, we obtain:

We now multiply this result by the exponent $g(x)=\frac{1}{x}$ to find the exponent of our base $e$:

We rearrange the expression to isolate the standard limit $\frac{\tan x}{x}$:

Since ${\lim }_{x\to 0}\frac{\tan x}{x}=1$ and ${\lim }_{x\to 0}\frac{1}{1-\tan x}=1$, the calculation simplifies to:

The original limit is equal to $e^L$. Substituting our calculated value of $L=2$, we arrive at the final result: