The value of l i m x → ∞ x + c o s x x + s i n x i s Options: A . &nbsp -1 B . &nbsp 0 C . &nbsp 1 D . &nbsp 2

Answer: Option C : C l i m x → ∞ x + c o s x x + s i n x [ Putting x = 1 h ; a s x → ∞ , h → 0 ] = l i m h → 0 1 h + c o s ( 1 h ) 1 h + s i n ( 1 h ) = l i m h → 0 1 + h c o s 1 h 1 + h c o s 1 h = 1 + 0 1 + 0 ⎡ ⎢ ⎢ ⎢ ⎣ ∵ − 1 ≤ s i n 1 h ≤ 1 a n d − 1 ≤ c o s 1 h ≤ 1 , ∴ w h e r e h → 0 , h c o s 1 h → 0 a n d h s i n 1 h → 0 ⎤ ⎥ ⎥ ⎥ ⎦ = 1

The value of limx→∞x+cos xx+sin xis

Discussion Forum : Limits Continuity And Differentiability

Question -

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

Options:

A . -1

B . 0

C . 1

D . 2

Answer: Option C
:
C

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

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Discussion Forum : Limits Continuity And Differentiability

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