Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f(x)=\{\begin{array}{ll}\{1+|\sin x|{\}}^{a/|\sin x|},& -\pi /6<x<0\\ b,& x=0\\ e^{\tan 2x/\tan 3x},& 0<x<\pi /6\end{array}$. Determine $a$ and $b$ such that $f(x)$ is continuous at $x=0$.

In the world of calculus, continuity is the ultimate test of structural integrity. Imagine a function as a road; if you reach $x=0$, you expect a smooth transition. If there is a gap, a jump, or a hole, the function is discontinuous.

Our mission is to find the values of $a$ and $b$ that ensure this bridge is perfectly connected at $x=0$.

The $1^{\infty }$ Beast

As we approach $x=0$ from the left, we encounter the expression $f(x)=\{1+|\sin x|{\}}^{\frac{a}{|\sin x|}}$. As $x\to 0^{-}$, the base $1+|\sin x|$ approaches $1$, while the exponent $\frac{a}{|\sin x|}$ approaches infinity.

We have arrived at the classic $1^{\infty }$ indeterminate form. We utilize the standard limit shortcut:

Here, $g(x)=|\sin x|$ and $h(x)=\frac{a}{|\sin x|}$. When we multiply these terms, the $|\sin x|$ components cancel out with surgical precision:

Thus, our Left Hand Limit is simply $e^a$.

The Trigonometric Dance

Now, we shift our gaze to the right side of the bridge, where $x>0$. Here, the function takes the form $f(x)=e^{\frac{\tan 2x}{\tan 3x}}$.

We need to evaluate the limit of the exponent: ${\lim }_{x\to 0^{+}}\frac{\tan 2x}{\tan 3x}$. Plugging in $x=0$ yields the $\frac{0}{0}$ indeterminate form.

We apply the standard limit ${\lim }_{\theta \to 0}\frac{\tan \theta }{\theta }=1$ by performing an algebraic maneuver:

As $x\to 0$, the first two terms approach $1$, and the $x$ variables cancel out. This leaves us with a limit of $\frac{2}{3}$. Therefore, our Right Hand Limit is $e^{\frac{2}{3}}$.

For the function to be continuous at $x=0$, the Left Hand Limit, the Right Hand Limit, and the function value $f(0)=b$ must all be equal. This gives us the condition:

By comparing the exponents, we find: $a=\frac{2}{3}$

By comparing the values, we find: $b=e^{\frac{2}{3}}$

You have successfully bridged the gap, solved the indeterminate forms, and mastered the continuity of this function. Keep this mindset—every complex problem is just a bridge waiting for you to connect the pieces.