$$\lim_{x\rightarrow \infty}\frac{x+\sin{x}}{x+2\sin{x}}$$ I have obtained different answers while using two different methods.
method1 : splitting the function into two parts $$\lim_{x\rightarrow \infty}\frac{x+2\sin{x}}{x+2\sin{x}} - \lim_{x\rightarrow \infty}\frac {\sin{x}}{x+2\sin{x}} $$ now, clearly the value of limits can be observed as 1-0=1
method2 : using L'Hospital's Rule
the given expression can be written as
$$\lim_{x\rightarrow \infty}\frac{1+\cos{x}}{1+2\cos{x}}$$
now,clearly we can see that limit to the above expression does not exist.
WHICH METHOD IS WRONG AND WHY??
The second one is wrong. L'Hopital's Rule states that, under certain conditions, if the limit $\lim_{x\to\infty}\frac{f'(x)}{g'(x)}$ exists, then $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\lim_{x\to\infty}\frac{f'(x)}{g'(x)}$. It says nothing about what happens when the limit $\lim_{x\to\infty}\frac{f'(x)}{g'(x)}$ does not exist.