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Limits And Derivatives

Multiple Choice Questions

$\lim_{x \to 0} \frac{1 - \cos^{3} (4 x)}{x^{2}} = ?$

$If f (x) = \sin (\cos (x)), then f' (x) = ?$

$\cos (\cos (x))$
$\sin (- \sin (x))$
$(\sin (x)) \cos (\cos (x))$
$(- \sin (x)) \cos (\cos (x))$

$What is the derivative of the function f (x) = e^{\tan (x)} + \ln (\sec (x)) - e^{\ln (x)} at x = \frac{π}{4} ?$

$\frac{e}{2}$
e
2e
4e

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

$\frac{1}{2}$
1
2
Limit does not exist

$What is \lim_{h \to 0} \frac{\sqrt{2 x + 3 h} - \sqrt{2 x}}{2 h} = ?$

$\frac{1}{2 \sqrt{2} x}$
$\frac{3}{\sqrt{2} x}$
$\frac{3}{2 \sqrt{2} x}$
$\frac{3}{2 \sqrt{2} x}$

6.
What is the value of $\log_{7} (\sqrt{7 \sqrt{7 \sqrt{7}}}) = ?$

$3 \log_{2} (7)$

$1 - 3 \log_{2} (7)$

$1 - 3 \log_{7} (2)$

$\frac{7}{8}$

$1 - 3 \log_{7} (2)$

$\sqrt{7 \sqrt{7 \sqrt{7}}} = 7^{\frac{2^{3} - 1}{2^{3}}} = 7^{\frac{7}{8}} ∴ \log_{7} (\log_{7} (\sqrt{7 \sqrt{7 \sqrt{7}}})) = \log_{7} {(\log_{7} (7))}^{\frac{7}{8}} = \log_{7} [\frac{7}{8} \log_{7} (7)] = \log_{7} (\frac{7}{8}) = \log_{7} (7) - \log_{7} (8) = 1 - \log_{7} {(2)}^{3} = 1 - 3 \log_{7} (2)$