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Limits and Derivatives

Multiple Choice Questions

101.

$\lim_{x \to 0} \frac{e^{x} - e^{\sin (x)}}{2 (x - \sin (x))}$

$- \frac{1}{2}$
$\frac{1}{2}$
1
$\frac{3}{2}$

102.

If f(x) = $\{\frac{\sin (1 + [x])}{[x]}, for [x] \neq 0 0, for [x] = 0\}$

where [x] denotes the greatest integer not exceeding x, then $\lim_{x \to 0^{-}} f (x)$ is equal to

103.

If f(x) = $\{x - 5, for x \leq 1 4 x^{2} - 9, for 1 < x < 2 3 x + 4, for x \geq 2\}$

then f'(2⁺) is equal to

104.

If 2x² - 3xy + y² + x + 2y - 8 = 0, then : $\frac{dy}{dx}$ is equal to

$\frac{3 y - 4 x - 1}{2 y - 3 x + 2}$
$\frac{3 y - 4 x + 1}{2 y + 3 x + 2}$
$\frac{3 y - 4 x + 1}{2 y - 3 x - 2}$
$\frac{3 y - 4 x + 1}{2 y + 3 x + 2}$

105.

$If z = \log (\tan (x) + \tan (y)), then \sin (2 x) \frac{\partial z}{\partial x} + \sin (2 y) \frac{\partial z}{\partial y} is equal to$

106.

$\lim_{x \to 0} \frac{(1 - e^{x}) \sin (x)}{x^{2} + x^{3}} = ?$

107.

$If R \to R is defined by f (x) = [x - 3] + |x - 4| for x \in R, then \lim_{x \to 3^{-}} f (x) = ?$

108.
$If x = a \{\cos (θ) + \log (\tan (\frac{θ}{2}))\} and y = asin (θ), then \frac{dy}{dx} = ?$

$\cot (θ)$

$\tan (θ)$

$\sin (θ)$

$\cos (θ)$

$\tan (θ)$

$Given that, x = a \{\cos (θ) + \log (\tan (\frac{θ}{2}))\} and y = asin (θ) On differentiating w . r . t . θ respectively, we get \frac{dy}{dθ} = a (- \sin (θ) + \frac{1}{\tan (\frac{θ}{2})} . \sec^{2} (\frac{θ}{2}) . \frac{1}{2}) = a (- \sin (θ) + \frac{1}{\sin (θ)}) = \frac{{acos}^{2} (θ)}{\sin (θ)} and \frac{dy}{dθ} = acos (θ) ∴ \frac{dy}{dx} = \frac{\frac{dy}{dθ}}{\frac{dx}{dθ}} = \frac{acos (θ)}{\frac{{acos}^{2} (θ)}{\sin (θ)}} = \tan (θ)$