lim x → ∞ √ x + sin x x − cos x = Options: A . &nbsp 0 B . &nbsp 1 C . &nbsp − 1 D . &nbsp does not exist

limx→∞√x+sinxx−cosx=

Question -

$lim x \to \infty \sqrt{\frac{x + sin x}{x - cos x}} =$

Options:

A .

0

B .

1

C .

- 1

D . does not exist

Answer: Option B
:
B

lim x \to \infty \sqrt{\frac{x + s i n x}{x - c o s x}}

= lim x \to \infty \frac{x^{\frac{1}{2}} \sqrt{1 + \frac{s i n x}{x}}}{x^{\frac{1}{2}} \sqrt{1 - \frac{c o s x}{x}}}

We know, that for any value of x, sinx and cosx will be [-1,1]
So,

= lim x \to \infty \frac{S i n x}{x} = 0

And

= lim x \to \infty \frac{C o s x}{x} = 0

= x^{\frac{1}{2}} \sqrt{1 + \frac{s i n x}{x}} \frac{\sqrt{1 + \frac{s i n x}{x}}}{\sqrt{1 + \frac{c o s x}{x}}}

= lim x \to \infty \frac{1}{1}

= 1

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