10 Question MCQ (including Assertion)

Assertion (A): If \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) both exist, then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) for all functions \( g(x) \).

Reason (R): The quotient law of limits holds provided the limit of the denominator in the limit process is non-zero.

Assertion (A): As the time interval \( \Delta t \to 0 \), the average velocity of a falling body with position \( s(t) = 4.9 t^2 \) over \( [t, t + \Delta t] \) approaches the instantaneous velocity \( 9.8 t \).

Reason (R): The instantaneous rate of change of displacement with respect to time is defined as the limiting value \( \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t} \).

Assertion (A): \( \lim_{x \to 5} (|x| - 5) = 0 \).

Reason (R): The absolute value function \( f(x) = |x| \) is continuous for all real numbers \( x \in \mathbb{R} \).

Assertion (A): \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).

Reason (R): For \( 0 < |x| < \frac{\pi}{2} \), the inequality \( \cos x < \frac{\sin x}{x} < 1 \) holds, and by Sandwich Theorem, as \( x \to 0 \), both lower and upper bounds approach 1.

Assertion (A): The slope of the tangent line to the curve \( y = x^2 - 2 \) at point \( x = 10 \) is equal to 20.

Reason (R): Geometrically, the value of the derivative \( f'(a) \) represents the slope \( \tan \psi \) of the tangent line to the curve \( y = f(x) \) at point \( (a, f(a)) \).