Key Concepts and Formulas

• Definition of the Derivative: The derivative of a function $f(x)$ at a point $x=a$, denoted by $f'(a)$, is defined as: $f'(a) = \mathop {\lim }\limits_{h \to 0} {{f(a+h) - f(a)} \over h}$ or equivalently, $f'(a) = \mathop {\lim }\limits_{x \to a} {{f(x) - f(a)} \over {x - a}}$
• L'Hôpital's Rule: If the limit of a quotient of two functions, $\mathop {\lim }\limits_{x \to a} {{g(x)} \over {h(x)}}$, results in an indeterminate form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then the limit is equal to the limit of the quotient of their derivatives, provided the latter limit exists: $\mathop {\lim }\limits_{x \to a} {{g(x)} \over {h(x)}} = \mathop {\lim }\limits_{x \to a} {{g'(x)} \over {h'(x)}}$
• Algebraic Manipulation of Limits: We can often rewrite expressions within limits to simplify them or to reveal a known form, such as the definition of the derivative.

Step-by-Step Solution

1. Identify the Limit Expression and Given Information: We are asked to find the limit: $L = \mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$ We are given that $f(x)$ is differentiable at $x=a$, with $f'(a) = 2$ and $f(a) = 4$.

2. Check for Indeterminate Form: As $x \to a$, the numerator approaches $a \cdot f(a) - a \cdot f(a) = a \cdot 4 - a \cdot 4 = 0$. As $x \to a$, the denominator approaches $a - a = 0$. Since the limit results in the indeterminate form $\frac{0}{0}$, we can use L'Hôpital's Rule or algebraic manipulation.

3. Method 1: Using L'Hôpital's Rule: Let $g(x) = xf(a) - af(x)$ and $h(x) = x - a$. Then $g'(x) = \frac{d}{dx}(xf(a) - af(x)) = f(a) - a f'(x)$. And $h'(x) = \frac{d}{dx}(x - a) = 1$. Applying L'Hôpital's Rule: $L = \mathop {\lim }\limits_{x \to a} {{g'(x)} \over {h'(x)}} = \mathop {\lim }\limits_{x \to a} {{f(a) - af'(x)} \over 1}$ Now, substitute $x=a$ into the expression, as the limit of $f'(x)$ as $x \to a$ is $f'(a)$ (since $f$ is differentiable at $a$, $f'$ is continuous at $a$ in the neighborhood of $a$). $L = f(a) - af'(a)$ Substitute the given values $f(a) = 4$ and $f'(a) = 2$: $L = 4 - a(2) = 4 - 2a$

4. Method 2: Algebraic Manipulation (Connecting to the Definition of Derivative): We want to manipulate the expression to resemble the definition of the derivative. The definition of the derivative is $\mathop {\lim }\limits_{x \to a} {{f(x) - f(a)} \over {x - a}} = f'(a)$. Let's rewrite the numerator by adding and subtracting a term that will help us isolate the derivative term. A common strategy is to add and subtract $af(a)$: $L = \mathop {\lim }\limits_{x \to a} {{xf(a) - af(a) + af(a) - af(x)} \over {x - a}}$ Group the terms: $L = \mathop {\lim }\limits_{x \to a} \left( {{xf(a) - af(a)} \over {x - a}} + {{af(a) - af(x)} \over {x - a}} \right)$ Factor out common terms in each numerator: $L = \mathop {\lim }\limits_{x \to a} \left( {{f(a)(x - a)} \over {x - a}} + {{a(f(a) - f(x))} \over {x - a}} \right)$ For $x \neq a$, we can cancel $(x-a)$ in the first term: $L = \mathop {\lim }\limits_{x \to a} \left( f(a) + a \cdot {{f(a) - f(x)} \over {x - a}} \right)$ We can split this into two limits: $L = \mathop {\lim }\limits_{x \to a} f(a) + \mathop {\lim }\limits_{x \to a} \left( a \cdot {{f(a) - f(x)} \over {x - a}} \right)$ The first limit is simply $f(a)$ because $f(a)$ is a constant with respect to $x$. $L = f(a) + a \cdot \mathop {\lim }\limits_{x \to a} \left( -{{f(x) - f(a)} \over {x - a}} \right)$ We can pull the constant $a$ out of the limit and move the negative sign: $L = f(a) - a \cdot \mathop {\lim }\limits_{x \to a} \left( {{f(x) - f(a)} \over {x - a}} \right)$ The limit term $\mathop {\lim }\limits_{x \to a} \left( {{f(x) - f(a)} \over {x - a}} \right)$ is the definition of $f'(a)$. $L = f(a) - a \cdot f'(a)$ Substitute the given values $f(a) = 4$ and $f'(a) = 2$: $L = 4 - a(2) = 4 - 2a$

Both methods yield the same result.

Common Mistakes & Tips

• Incorrectly Applying L'Hôpital's Rule: Ensure that the limit indeed results in an indeterminate form ($\frac{0}{0}$ or $\frac{\infty}{\infty}$) before applying L'Hôpital's Rule.
• Algebraic Errors: Be careful with signs and factoring when manipulating the expression algebraically. A common mistake is misplacing the negative sign when rearranging terms to match the derivative definition.
• Recognizing the Derivative Definition: The expression $\mathop {\lim }\limits_{x \to a} {{f(x) - f(a)} \over {x - a}}$ is fundamental. If you can rewrite the given limit to include this form, it often simplifies the problem.

Summary

The problem asks for the evaluation of a limit involving a differentiable function and its values at a point. We first confirm that the limit is in an indeterminate form. We then employ two valid methods: L'Hôpital's Rule, which involves taking derivatives of the numerator and denominator, and algebraic manipulation, which rewrites the expression to explicitly use the definition of the derivative. Both methods lead to the expression $f(a) - af'(a)$. Substituting the given values $f(a)=4$ and $f'(a)=2$, we find the limit to be $4 - 2a$.

The final answer is $\boxed{4 - 2a}$.