The value of x-limx-cos x-x-sin xis

X→∞limx+cos x/x+sin x [ Puttingx=1/h;as x→∞,h →0 ] =h→ 0lim1/h+cos ( 1/h )/1/h+sin ( 1/h ) =h→ 0lim1+h cos 1/h/1+h cos 1/h=1+0/1+0 [ ∵ 1≤ sin 1/h≤ 1 and ...

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Byju's Answer

Standard XIII

Mathematics

Sandwich Theorem

The value of ...

Question

$The value of l i m x \to \infty \frac{x + c o s x}{x + s i n x} i s$

-1

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Solution

The correct option is C
1

$l i m x \to \infty \frac{x + c o s x}{x + s i n x} [Putting x = \frac{1}{h}; a s x \to \infty, h \to 0] = l i m h \to 0 \frac{\frac{1}{h} + c o s (\frac{1}{h})}{\frac{1}{h} + s i n (\frac{1}{h})} = l i m h \to 0 \frac{1 + h c o s \frac{1}{h}}{1 + h c o s \frac{1}{h}} = \frac{1 + 0}{1 + 0} ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ∵ - 1 \leq s i n \frac{1}{h} \leq 1 a n d - 1 \leq c o s \frac{1}{h} \leq 1, ∴ w h e r e h \to 0, h c o s \frac{1}{h} \to 0 a n d h s i n \frac{1}{h} \to 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ = 1$

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$The value of l i m x \to \infty [\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}] is .$

l i m_{x \to \infty} c o s (\sqrt{x + 1}) - c o s (\sqrt{x})

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The value of limx→∞x+cos xx+sin xis

The correct option is C 1 limx→∞x+cos xx+sin x[Puttingx=1h;as x→∞,h →0]=limh→ 01h+cos(1h)1h+sin(1h)=limh→ 01+h cos 1h1+h cos 1h=1+01+0⎡⎢ ⎢ ⎢⎣∵−1≤ sin 1h≤1 and −1≤ cos1h≤1,∴ where h→0,hcos 1h→0 and hsin1h→0⎤⎥ ⎥ ⎥⎦=1

$The value of l i m x \to \infty \frac{x + c o s x}{x + s i n x} i s$