The limiting value of cos x1 - sin x as x - 0 isA- 0 B- 1 - eC- 1D- e

The correct option is C 1 Put cos x = 1 + y as; x →0; y →0 Hence lim_x→0 (cos x)^1/sin x=lim_y→0[(1+y)^1/y]^y/sin x e^x→ 0limcosx 1/sin x =e^x→ 0lim( tanx/2)= ...

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Standard XII

Mathematics

Indeterminate Forms

The limiting ...

Question

$The limiting value of {(c o s x)}^{1 / s i n x} as~x \to 0 i s$

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Solution

The correct option is A
1

Put cos x = 1 + y as; $x \to 0$ ; $y \to 0$

Hence

${lim}_{x \to 0} (c o s x)^{1 / s i n x} = {lim}_{y \to 0} [(1 + y)^{1 / y}]^{y / s i n x} e^{l i m x \to 0 \frac{c o s x - 1}{s i n x}} = e^{l i m x \to 0 (- t a n \frac{x}{2})} = e^{0} = 1$

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Similar questions

$The limiting value of {(c o s x)}^{1 / s i n x} as x \to 0 i s$

Q. If the function

f (x) = \{\begin{cases} {(\cos x)}^{1 / x}, & x \neq 0 \\ k, & x = 0 \end{cases}

is continuous at x = 0, then the value of k is
(a) 0
(b) 1
(c) −1
(d) e.

l i m i t_{x \to 0} \frac{e^{x} - 1}{s i n x}

$lim x \to 0 \frac{l o g_{e} (1 + x)}{x} =$

Q. If

f (x)

is an odd function and

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exists then

f^{'} (e) - f^{'} (- e)

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The limiting value of(cos x)1/sin xas~x→0 is

The correct option is A 1 Put cos x = 1 + y as; x→0; y→0 Hence limx→0 (cos x)1/sin x=limy→0[(1+y)1/y]y/sin xelimx→0cosx−1sin x=elimx→0(−tanx2)=e0=1

$The limiting value of {(c o s x)}^{1 / s i n x} as~x \to 0 i s$

The correct option is A
1

Put cos x = 1 + y as; $x \to 0$ ; $y \to 0$

Hence

${lim}_{x \to 0} (c o s x)^{1 / s i n x} = {lim}_{y \to 0} [(1 + y)^{1 / y}]^{y / s i n x} e^{l i m x \to 0 \frac{c o s x - 1}{s i n x}} = e^{l i m x \to 0 (- t a n \frac{x}{2})} = e^{0} = 1$