CUET PG 2026 — DSA PYQ

To find the asymptotic relationship between $f(n)$ and $g(n)$, evaluate the limit of their ratio as $n\to \infty$:

$$\underset{n\to \infty }{\lim }\frac{f(n)}{g(n)}=\underset{n\to \infty }{\lim }\frac{5n^2+3n\log n}{n^2}$$

Dividing each term in the numerator by $n^2$:

$$\underset{n\to \infty }{\lim }\left(\frac{5n^2}{n^2}+\frac{3n\log n}{n^2}\right)=\underset{n\to \infty }{\lim }\left(5+\frac{3\log n}{n}\right)$$

As $n\to \infty$, $\frac{\log n}{n}\to 0$:

$$5+0=5$$

Since the limit is a positive constant (0 < 5 < \infty), $f(n)$ and $g(n)$ have the same order of growth.

By definition:

• $f(n)=\Theta (g(n))$

Correct Answer: (C) $f(n)=\Theta (g(n))$