First slide

Continuity

Question

$If f (x) = \int \frac{x + \sin x}{x + \cos x} d x, then lim_{x \to \infty} f (x) =$

Moderate

A

0
B

1
C

$\infty$
D

None of these

Solution

$\begin{array}{l} W e h a v e f (x) = \int \frac{x + \sin x}{x + \cos x} d x \\ \Rightarrow f (x) = \frac{x + \sin x}{x + \cos x} \\ \Rightarrow \lim_{x \to \infty} f (x) = \lim_{x \to \infty} \frac{1 + \frac{\sin x}{x}}{1 + \frac{\cos x}{x}} = \frac{1 + 0}{1 + 0} = 1 \\ [∴ \lim_{x \to \infty} \frac{1}{x} = 0 a n d |\sin x| \leq 1, |\cos x| \leq 1 f o r a l l x] \\ H e n c e (2) i s t h e c o r r e c t a n s w e r . \end{array}$

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