IGDTUW 2026 — Mathematics PYQ

Step 1: Understand the Behavior at Infinity

Let the limit of the function as $x$ approaches infinity be a constant $L$. That is:

$$\underset{x\to \infty }{\lim }f(x)=L$$

Since $x\to \infty$, the terms $x+1$ and $x+2$ will also approach infinity. Therefore, their limits will also be equal to $L$:

$$\underset{x\to \infty }{\lim }f(x+1)=L$$

$$\underset{x\to \infty }{\lim }f(x+2)=L$$

Step 2: Set up the Equation

Given the functional equation:

$$f(x)=\frac{1}{3}\left\{f(x+1)+\frac{5}{f(x+2)}\right\}$$

Taking the limit as $x\to \infty$ on both sides:

$$L=\frac{1}{3}\left(L+\frac{5}{L}\right)$$

Step 3: Solve for $L$

Multiply both sides by 3 to clear the fraction:

$$3L=L+\frac{5}{L}$$

Subtract $L$ from both sides:

$$2L=\frac{5}{L}$$

Multiply both sides by $L$:

$$2L^2=5$$

$$L^2=\frac{5}{2}$$

Taking the square root on both sides:

$$L=\pm \sqrt{\frac{5}{2}}$$

Step 4: Final Conclusion

Since the problem states that f(x) > 0 for all $x\in R$, the limit $L$ must also be positive.

Therefore:

$$L=\sqrt{\frac{5}{2}}$$

Correct Answer:

(b) $\sqrt{\frac{5}{2}}$