Consider the function $$ f(x)=\lim _{n \rightarrow \infty} \frac{\log _{e}(2+x)-x^{2 n} \sin x}{1+x^{2 n}}, x>0 $$ Is $f(x)$ continuous at $x=1 ?$ Justify your answer.
Applying basic theorem of continuity,calculating the function value $\lim_{h\to1^{+}} f(h+1)$ and $\lim_{h\to1^{-}} f(1-h)$, and then $f(1)$ ..I am getting this as non continous. Am I wrong?
Hint: Recall that $x^{k} \to 0$ as $k \to \infty$ if $0<x<1$ and $x^{k} \to \infty$ as $k \to \infty$ if $x>1$. Hence. $f(x)=\log_2(2+x)$ for $x <1$ and $f(x)=-\sin x $ for $x >1$. The left hand limit is $\log_e 3$ and the right hand limit is $-\sin 1$ These are not equal so $f$ is not continuous at $1$.