Animated Solution for Mathematics - Limits, Continuity and Differentiability: Use the formula ${\lim }_{x\to 0}\frac{a^x-1}{x}=\ln a$ to find ${\lim }_{x\to 0}\frac{2^x-1}{(1+x{)}^{1/2}-1}$

The problem asks us to evaluate the limit:

The first rule of any limit problem is to test the waters with direct substitution. When we plug in $x=0$, the numerator becomes $2^0-1=0$, and the denominator becomes $(1+0{)}^{1/2}-1=0$.

We are staring at a $\frac{0}{0}$ indeterminate form. This is not a dead end; it is a sign that the function is hiding its true value behind a veil of algebraic complexity.

To break through, we utilize the standard limit formula:

Our numerator, $2^x-1$, is almost perfect, but it lacks an $x$ in the denominator. To fix this without altering the expression, we perform a surgical strike: we divide both the numerator and the denominator by $x$.

This transforms our expression into:

By the properties of limits, we can now treat the numerator and the denominator as two separate entities:

The first part is now a direct application of our standard formula. With $a=2$, the limit simplifies beautifully:

Just like that, we have conquered the exponential part of the problem. It is a moment of pure mathematical satisfaction when a complex term collapses into a simple constant.

Now, we turn our attention to the second part: ${\lim }_{x\to 0}\frac{x}{(1+x{)}^{1/2}-1}$. If we try to evaluate this directly, we still face the $\frac{0}{0}$ problem.

Whenever you see a square root causing this issue, your best friend is rationalization. We multiply the numerator and the denominator by the conjugate, $(\sqrt{1+x}+1)$.

The denominator becomes $(\sqrt{1+x}-1)(\sqrt{1+x}+1)$, which simplifies to $(1+x)-1$, leaving us with just $x$. The expression now looks like:

The $x$ terms cancel out, leaving us with ${\lim }_{x\to 0}(\sqrt{1+x}+1)$. Substituting $x=0$ gives us $\sqrt{1}+1=2$.

We have our two pieces: the first limit is $\ln 2$ and the second is $2$. Multiplying them together, we arrive at our final answer:

This journey shows that even the most daunting limits are just puzzles waiting to be solved with the right tools. Keep practicing, stay curious, and remember that every 'indeterminate' form is just an opportunity to reveal the elegance hidden beneath.