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Limits and Derivatives

Multiple Choice Questions

101.

$\lim_{x \to \infty} {(\frac{x + 5}{x + 2})}^{x + 3}$ equals

e
e²
e³
e⁵

102.

$\lim_{x \to 0} \frac{\tan (x) - \sin (x)}{x^{2}} = ?$

0
1
$\frac{1}{2}$
$- \frac{1}{2}$

103.

$If f (x) = |x| + |\sin (x)| for x \in (- \frac{π}{2}, \frac{π}{2}), then its left hand derivative at x = 0 is$

104.

$If u = u (x, y) = \sin (y + ax) - {(y + ax)}^{2}, then it implies$

$u_{xx} = a^{2} . u_{yy}$
$u_{yy} = a^{2} u_{xx}$
$u_{xx} = - a^{2} . u_{yy}$
$u_{yy} = - a^{2} u_{xx}$

105.

$\lim_{x \to \infty} {(\frac{x + 6}{x + 1})}^{x + 4} = ?$

e⁴
e⁶
e⁵
e

106.
The coordinates of the point P on the curve $x = a (θ + \sin (θ)), y = a (1 - \cos (θ)),$ where the tangent is inclined at an angle $\frac{π}{4}$ to x-axis, are

$[a (\frac{π}{4} - 1), a]$

$[a (\frac{π}{2} + 1), a]$

$[a (\frac{π}{2}), a]$

(a, a)

$[a (\frac{π}{2} + 1), a]$

$Given coordinate is x = a (θ + \sin (θ)), \frac{dy}{dθ} = a (0 + \sin (θ)) On differentiating w . r . t . x, we get \frac{dx}{dθ} = a (1 + \cos (θ)), \frac{dy}{dθ} = a (0 + \sin (θ)) ∴ \frac{dy}{dθ} = \frac{\frac{dy}{dθ}}{\frac{dx}{dθ}} = \frac{asin (θ)}{a (1 + \cos (θ))} = \frac{2 \sin (\frac{θ}{2}) \cos (\frac{θ}{2})}{2 \cos^{2} (\frac{θ}{2})} = \tan (\frac{θ}{2}) \tan (\frac{π}{4}) = \tan (\frac{θ}{2}) (∴ \frac{dy}{dθ} = \tan (x)) \Rightarrow \frac{π}{4} = \frac{θ}{2} \Rightarrow θ = \frac{π}{2}$

$∴ Coordinate of P [a (\frac{π}{2} + \sin (\frac{π}{2})), a (1 - \cos (\frac{π}{2}))] = P [a (\frac{π}{2} + 1), a]$

107.

$If f (x) = {xtan}^{- 1} (x), then \lim_{x \to 1} \frac{f (x) - f (1)}{x - 1} = ?$

$\frac{π + 3}{4}$
$\frac{π}{4}$
$\frac{π + 1}{4}$
$\frac{π + 2}{4}$

108.

$If g (x) = \frac{x}{|x|} for x > 2, then \lim_{x \to 2} \frac{g (x) - g (2)}{x - 2} = ?$

109.

$\lim_{x \to \frac{π}{2}} (\frac{2 x - π}{\cos (x)}) = ?$

110.

$\lim_{x \to 0} \frac{6^{x} - 3^{x} - 2^{x} + 1}{x^{2}} = ?$

$(\log_{e} (2)) \log_{e} (3)$
$\log_{e} (5)$
$\log_{e} (6)$
0

Previous Year Papers

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Test Yourself

Multiple Choice Questions

106. The coordinates of the point P on the curve x = aθ + sinθ, y = a1 - cosθ, where the tangent is inclined at an angle π4 to x-axis, are aπ4 - 1, a aπ2 + 1, a aπ2, a (a, a)

106.
The coordinates of the point P on the curve $x = a (θ + \sin (θ)), y = a (1 - \cos (θ)),$ where the tangent is inclined at an angle $\frac{π}{4}$ to x-axis, are

$[a (\frac{π}{4} - 1), a]$

$[a (\frac{π}{2} + 1), a]$

$[a (\frac{π}{2}), a]$

(a, a)