Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to 0}{\left(\tan \left(\frac{\pi }{4}+x\right)\right)}^{\frac{1}{x}}$ is equal to :

A Journey into Limits

Welcome, fellow traveler of the mathematical landscape. Today, we are going to dismantle a classic problem that often strikes fear into the hearts of students: the indeterminate form $1^{\infty }$.

When you first look at the limit it might seem daunting. But I want you to take a deep breath; we are not just solving an equation, we are peeling back the layers of a geometric reality.

Identifying the Trap

In the world of limits, our first instinct should always be direct substitution. Imagine standing at the point $x=0$.

As we approach this point, what happens to our base? We have $\tan (\frac{\pi }{4}+0)$, which is simply $\tan (\frac{\pi }{4})=1$.

Now, look at the exponent: $\frac{1}{x}$. As $x$ shrinks toward zero, this fraction explodes toward infinity. We have arrived at the $1^{\infty }$ form, which is a signal that we need to dig deeper.

The Elegant Shortcut

Instead of getting lost in the weeds of logarithms, we have a powerful tool in our arsenal. Whenever you see a limit of the form ${\lim }_{x\to a}[f(x){]}^{g(x)}$ resulting in $1^{\infty }$, you can immediately rewrite it as:

Think of this as a bridge that carries us from the exponential world into the world of simple multiplication. Here, our $f(x)=\tan (\frac{\pi }{4}+x)$ and our $g(x)=\frac{1}{x}$.

Our mission is now to evaluate the limit of the exponent:

Trigonometric Surgery

Now, we need to perform some surgery on that tangent term. We recall the identity:

By setting $A=\frac{\pi }{4}$ and $B=x$, and knowing that $\tan (\frac{\pi }{4})=1$, we get:

Watch closely as we substitute this back into our exponent expression. We now have:

To combine these terms, we find a common denominator:

When we distribute that negative sign, the ones vanish, leaving us with:

The Final Tally

We are almost at the finish line. Our exponent expression has simplified to:

Let us group these terms strategically:

Why did we do this? Because we know the fundamental standard limit ${\lim }_{x\to 0}\frac{\tan x}{x}=1$.

As $x$ approaches zero, the first part becomes $1$, and the second part, $\frac{1}{1-\tan x}$, becomes $\frac{1}{1-0}=1$. Multiplying these by our constant $2$, the entire exponent limit evaluates to $2$.

Finally, we return to our base $e$. The limit we sought is $e^2$. You see? What looked like a terrifying mountain was just a series of small, logical steps.