Let f : R → R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of lim x→2 (x²f(2) - 4

Q: Wait, why do we treat $f(2)$ as a constant when differentiating the numerator?

That is a brilliant question! Remember that $f(2)$ is just a specific value of the function at $x=2$. It is a constant number, not a variable. When we differentiate with respect to $x$, any constant multiplier remains unchanged, just like the $4$ in $4x^2$.

Q: Can I use the definition of the derivative instead of L'Hospital's Rule?

Absolutely! You could rewrite the limit in terms of the definition of the derivative, $f'(2) = \lim_{x \to 2} \frac{f(x) - f(2)}{x - 2}$. It is a more fundamental approach, but L'Hospital's Rule is often faster and more direct for these types of problems. Both paths lead to the same beautiful destination.

Q: What if the denominator wasn't just $x-2$? Would L'Hospital's still work?

Yes, L'Hospital's Rule is robust! As long as the limit is in an indeterminate form like $0/0$ or $\infty/\infty$, you can apply it. The key is to ensure both the numerator and denominator are differentiable at the point of interest.

Question

Let $f : \mathbb{R} 	o \mathbb{R}$ be a function such that $f(2) = 4$ and $f'(2) = 1$. Then, the value of $\lim_{x 	o 2} \frac{x^2 f(2) - 4 f(x)}{x - 2}$ is equal to :

Options:

(A) 4

(B) 8

(C) 16

(D) 12

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler on the journey of calculus! Today, we are going to unravel a problem that might look intimidating at first glance, but beneath its surface lies a beautiful, elegant structure.

We are given a function $f(x)$ with two precious pieces of information: $f(2) = 4$ and $f'(2) = 1$. Think of these as our compass and map.

We are asked to evaluate the limit:

$$L = \lim_{x 	o 2} \frac{x^2 f(2) - 4 f(x)}{x - 2}$$

Before we start calculating, let's pause and appreciate the geometry. The derivative $f'(2) = 1$ tells us that at the point where $x=2$, the slope of the tangent line to our function is exactly $1$. This is the heartbeat of the problem.

### The Indeterminate Trap

Now, the most common mistake students make is to jump straight into complex algebraic manipulation. But wait! The first rule of limits is always to check the form.

Let's substitute $x = 2$ into our expression. The numerator becomes:

$$2^2 f(2) - 4 f(2) = 4(4) - 4(4) = 0$$

The denominator becomes $2 - 2 = 0$. We have a $0/0$ indeterminate form! This is not a dead end; it is a green light. It tells us that the function is well-behaved and that we have a powerful tool at our disposal.

### The Heroic Intervention

When we see $0/0$, we call upon our superhero: **L'Hospital's Rule**. This rule is a favorite in JEE because it elegantly resolves these indeterminate forms.

It states that for a limit of the form $0/0$, we can differentiate the numerator and the denominator separately with respect to $x$. So, let's differentiate the numerator $x^2 f(2) - 4 f(x)$.

Remember, $f(2)$ is just a constant! So, the derivative of $x^2 f(2)$ is $2x f(2)$, and the derivative of $-4 f(x)$ is $-4 f'(x)$. The denominator $x - 2$ is even simpler; its derivative is just $1$.

### The Final Victory

Now, our limit expression has transformed into:

$$\lim_{x 	o 2} \frac{2x f(2) - 4 f'(x)}{1}$$

This is much friendlier! We can now substitute $x = 2$ directly.

We get $2(2) f(2) - 4 f'(2)$. Using our given values $f(2) = 4$ and $f'(2) = 1$, we calculate:

$$4(4) - 4(1) = 16 - 4 = 12$$

And there it is! The final answer is **12**. You have successfully navigated the trap, applied the rule, and arrived at the solution. Keep this confidence with you; every complex problem is just a series of simple, logical steps waiting for you to uncover them.

Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f : R \to R$ be a function such that $f (2) = 4$ and $f^{'} (2) = 1$ . Then, the value of $lim_{x \to 2} \frac{x ^{2} f ( 2 ) - 4 f ( x )}{x - 2}$ is equal to :

Select Answer:

Visualized Solution

Given Information

Visualizing the Tangent

Analyzing the Limit Form

Checking the Numerator

Checking the Denominator

Applying L'Hospital's Rule

Differentiating the Numerator

Differentiating the Denominator

The New Limit Expression

Final Substitution

Final Answer

The Sigma Insight: Evaluation of Limits & L'Hopital's Rule

Analyzing the Setup

The Indeterminate Trap

The Heroic Intervention

The Final Victory

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