Q.Evaluate $\lim_{x \to a} \dfrac{(2 + x)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}}}{x - a}$.

This limit directly represents the definition of the derivative of a function at a point.

Concept: Definition of Derivative

1. The given limit $\lim_{x \to a} \dfrac{(2 + x)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}}}{x - a}$ matches the form $f'(a) = \lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}$.
2. By comparing, we identify the function as $f(x) = (2 + x)^{\frac{5}{2}}$. …

When evaluating limits, our first step is always to try direct substitution. If this gives a finite, defined value, that's our answer. However, often we encounter "indeterminate forms" like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. These forms tell us that the limit might exist, but we need to do more work to find it.

This particular problem presents a classic indeterminate form that directly relates to the fundamental definition of a derivative. Understanding this connection is key to solving it efficiently and conceptually.

Let's break down the solution.

1. Identify the Indeterminate Form

   First, we substitute $x=a$ into the expression:

   Numerator: $(2 + a)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}} = 0$

   Denominator: $a - a = 0$

   Since we get $\frac{0}{0}$, this is an indeterminate form, meaning we cannot determine the limit by direct substitution alone. We need to use other techniques.

2. Recognize the Definition of a Derivative

   The structure of the given limit is highly specific. Recall the definition of the derivative of a function $f(x)$ at a point $x=a$:

   $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

   Let's compare this definition with our given limit:

If we let $f(x) = (2+x)^{\frac{5}{2}}$, then $f(a) = (2+a)^{\frac{5}{2}}$.
The expression perfectly matches the definition of $f'(a)$. Therefore, to evaluate the limit, we simply need to find the derivative of $f(x)$ and then substitute $x=a$.

3. Differentiate the Function

Our function is $f(x) = (2+x)^{\frac{5}{2}}$.

To differentiate this, we use the power rule combined with the chain rule.

The power rule states that $\frac{d}{dx}(u^n) = n u^{n-1} \frac{du}{dx}$.

Here, $u = (2+x)$ and $n = \frac{5}{2}$.

The derivative of $u = (2+x)$ with respect to $x$ is $\frac{du}{dx} = \frac{d}{dx}(2+x) = 1$.

So, applying the power rule:

4. Evaluate the Derivative at $x=a$ Now that we have $f'(x)$, we substitute $x=a$ to find the value of the limit:

This is the value of the limit.