Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1) = 2 and lim

Q: Wait, why do we treat $x$ as a constant when differentiating with respect to $t$?

Great question! In the limit $\lim_{t \to x}$, $x$ is the fixed point toward which $t$ is approaching. Because $x$ does not change as $t$ varies, it behaves exactly like a constant number (like 5 or $\pi$) during the differentiation process. This is the secret to unlocking the differential equation hidden inside the limit.

Q: Why can't we just use L'Hôpital's Rule twice if the first derivative doesn't look like a standard ODE?

You could, but you don't need to! Once you differentiate once and substitute $t=x$, you arrive at a linear relationship between $f(x)$ and $f'(x)$. That is already the differential equation you need. Adding more layers of differentiation would only complicate the algebra unnecessarily.

Q: How do I know if my Integrating Factor is correct?

A quick sanity check: if you multiply your original ODE by your calculated $I.F.$, the left-hand side should collapse perfectly into the derivative of a product, i.e., $\frac{d}{dx}[f(x) \cdot I.F.]$. If it doesn't, re-check your $P(x)$ term!

Q: Is it possible to solve this without differential equations?

While some problems allow for clever substitutions, this specific structure—a limit defining a relationship between a function and its derivative—is the textbook definition of a differential equation. Embracing the ODE approach is the most robust and elegant path to the solution.

Q: I got $C = 13/11$, but the options look slightly different. Did I mess up?

Take a deep breath! Check your algebra on the constant $C$. You found $C = 2 - 9/11$. Since $2 = 22/11$, then $22/11 - 9/11$ is indeed $13/11$. If your final expression matches $f(x) = \frac{9}{11x} + \frac{13}{11}x^{10}$, you are absolutely correct. Always trust your derivation!

Question

Let $f(x)$ be a continuously differentiable function on the interval $(0,\infty)$ such that $f(1) = 2$ and $\lim_{t	o x} \frac{t^{10}f(x) - x^{10}f(t)}{t^{9} - x^{9}} = 1$ for each $x > 0$. Then, for all $x > 0$, $f(x)$ is equal to

Options:

(A) $\frac{31}{11x} - \frac{9}{11}x^{10}$

(B) $\frac{9}{11x} + \frac{31}{11}x^{10}$

(C) $-\frac{9}{11x} + \frac{31}{11}x^{10}$

(D) $\frac{13}{11x} + \frac{9}{11}x^{10}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the edge of a mathematical landscape. You are presented with a limit, a seemingly static expression:

$$\lim_{t	o x} \frac{t^{10}f(x) - x^{10}f(t)}{t^{9} - x^{9}} = 1$$

At first glance, it looks like a wall. But in the world of JEE Advanced, a limit is never just a wall; it is a doorway. When you see $t 	o x$ and a fraction that evaluates to $0/0$, you are looking at the birth of a differential equation.

### Unmasking the Derivative

We begin by applying **L'Hôpital's Rule**. When we differentiate with respect to $t$, $x$ is treated as a constant.

The numerator, $\frac{d}{dt} [t^{10}f(x) - x^{10}f(t)]$, becomes $10t^9 f(x) - x^{10}f'(t)$. The denominator, $\frac{d}{dt} [t^9 - x^9]$, simplifies to $9t^8$.

Now, we let $t$ approach $x$. The expression transforms into:

$$\frac{10x^9 f(x) - x^{10}f'(x)}{9x^8} = 1$$

Look at the elegance of the cancellation. By dividing the numerator terms by $x^8$, we strip away the complexity to reveal:

$$\frac{10x f(x) - x^2 f'(x)}{9} = 1$$

Rearranging this, we find the heartbeat of the problem: $x^2 f'(x) - 10x f(x) = -9$. Dividing by $x^2$, we arrive at the standard form of a **first-order linear differential equation**:

$$f'(x) - \frac{10}{x} f(x) = -\frac{9}{x^2}$$

### The Integrating Factor

We identify $P(x) = -\frac{10}{x}$ and $Q(x) = -\frac{9}{x^2}$. To solve this, we calculate the **Integrating Factor ($I.F.$)**:

$$I.F. = e^{\int P(x) dx} = e^{\int -\frac{10}{x} dx} = e^{-10 \ln x} = x^{-10}$$

This is the moment where the problem begins to resolve. When we multiply our ODE by $x^{-10}$, the left side becomes the derivative of a product:

$$\frac{d}{dx} [f(x) \cdot x^{-10}] = -9x^{-12}$$

### The Final Integration

We are in the home stretch. We integrate both sides with respect to $x$:

$$\int \frac{d}{dx} [f(x) \cdot x^{-10}] dx = \int -9x^{-12} dx$$

The integral of $-9x^{-12}$ is $-9 \cdot \frac{x^{-11}}{-11} + C$, which simplifies to $\frac{9}{11x^{11}} + C$. Multiplying through by $x^{10}$ to isolate $f(x)$, we get:

$$f(x) = \frac{9}{11x} + C x^{10}$$

Finally, we use our initial condition, $f(1) = 2$. Substituting these values, we find $2 = \frac{9}{11} + C$, which leads us to $C = \frac{13}{11}$.

### Conclusion

We have arrived at the destination:

$$f(x) = \frac{9}{11x} + \frac{13}{11}x^{10}$$

You didn't just solve a problem; you navigated a sequence of logical transformations. You took a limit, turned it into a differential equation, used an integrating factor to tame it, and applied boundary conditions to find the truth. This is the essence of mathematics—finding the hidden order within the chaos.

Select Answer:

Visualized Solution

Analyzing the Limit

Applying L'Hôpital's Rule

Evaluating the Limit

Simplifying to ODE Form

Integrating Factor Setup

Computing Integrating Factor

Solving the ODE

Integrating Both Sides

Using Initial Condition

Final Function

The Sigma Insight: Linear Differential Equations

Analyzing the Setup

Unmasking the Derivative

The Integrating Factor

The Final Integration

Conclusion

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