Previous gate question

Question

$\lim_{x\rightarrow infinity } \frac{x+sinx}{x}$

Answer 1

$\large lim_{x \rightarrow \infty}\; \frac{x + \sin\;x}{x} = 1 + lim_{x \rightarrow \infty}\;\frac{\sin\;x}{x}$

$\large lim_{x \rightarrow \infty}\;\frac{\sin\;x}{x} = 0$

So, $\large lim_{x \rightarrow \infty}\; \frac{x + \sin\;x}{x} = 1 +0 = 1$

Answer 2

lim_x->infinity 1 +lim_x->infinitysinx/x   (by property lim(f+g)=limf+limg)

1+0  (since on putting infinity in numerator we get any finite number and putting in denominator we get infinity and dividing finite with infinite we get 0)

hence answer is 1

Answer 3

it will give infinity by infinity which is a indertimnant form so aplly l hospital rule we will get

lim x→infinity 1 = 1          -----answer is 1