JAMIA 2026 — Mathematics PYQ

Step 1: Divide numerator and denominator by $x^2$

Divide each term individually:

$$L=\underset{x\to \infty }{\lim }\frac{\frac{5x^2}{x^2}+\frac{7}{x^2}}{\frac{2x^2}{x^2}+\frac{3x}{x^2}}$$

Simplify the terms inside the expression:

$$L=\underset{x\to \infty }{\lim }\frac{5+\frac{7}{x^2}}{2+\frac{3}{x}}$$

Step 2: Apply the limit $x\to \infty$

As $x$ grows infinitely large ($x\to \infty$), any fraction with a constant numerator and $x$ in the denominator approaches $0$:

• ${\lim }_{x\to \infty }\frac{7}{x^2}=0$

• ${\lim }_{x\to \infty }\frac{3}{x}=0$

Substituting these values into our expression gives:

$$L=\frac{5+0}{2+0}$$

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

Shortcut Rule for Rational Limits at Infinity:

When evaluating ${\lim }_{x\to \infty }\frac{P(x)}{Q(x)}$ where the degree of the numerator polynomial equals the degree of the denominator polynomial, the limit is simply the ratio of their leading coefficients.

Here, the leading coefficient of the numerator is 5 and the denominator is 2. Therefore, the answer is directly $\frac{5}{2}$.

Correct Answer:

(a) 5/2