Important Questions for Class 11 Maths Chapter 12 Limits and Derivatives
Limits
Question 1:
Evaluate: \( \lim_{x \to 0} (\csc x – \cot x) \)
Question 2:
Evaluate: \( \lim_{x \to \frac{\pi}{6}} \frac{\sqrt{3} \sin x – \cos x}{\frac{\pi}{6} – x} \)
Continuity
Question 1:
Let f(x) = \(\begin{cases}\frac{k \sin 2x}{x}, & \text{if } x < 0 \\x + 8, & \text{if } x \geq 0\end{cases} \)
Find K, \(\lim_{x \to 0} f(x) \)
Question 2:
Find whether the function f (x) is continuous or discontinuous at the given point
f(x) = \(\begin{cases}\frac{1-cos 2x}{x^2}, & \text{if } x \neq0 \\5, & \text{if } x = 0\end{cases} \) at x = 0.
Answer
\(
f(x) =
\begin{cases}
\frac{1 – \cos(2x)}{x^2}, & \text{if } x \neq 0 \\5, & \text{if } x = 0
\end{cases}
\)
$$
\lim_{x \to 0} \frac{1 – \cos(2x)}{x^2}
= \lim_{x \to 0} \frac{2\sin^2(x)}{x^2}
= 2
$$
$$
f(0) = 5
$$
$$
\text{Since } \lim_{x \to 0} f(x) = 2 \neq 5 = f(0), \text{ the function is discontinuous at } x = 0.
$$
Derivatives
Question 1:
If \(
y = \frac{2x + 5}{3x – 4}
\), then find \( \frac{dy}{dx} \)
Question 2:
