Prove that the limit $$\lim_{x \to \infty} (x/(x+1))(\sin(x^2)) = \lim_{x \to \infty} \frac{x \sin x^2 }{x+1}$$ does not exist as x approaches positive infinity
[Math] Real Analysis Prove that Limit Does not Exist
limitsreal-analysis
limitsreal-analysis
Prove that the limit $$\lim_{x \to \infty} (x/(x+1))(\sin(x^2)) = \lim_{x \to \infty} \frac{x \sin x^2 }{x+1}$$ does not exist as x approaches positive infinity
Best Answer
It is sufficient to choose two sequences $\left\lbrace {x_n^{(1)}}\right\rbrace$ and $\left\lbrace {x_n^{(2)}}\right\rbrace$ such that $$\lim\limits_{n \rightarrow \infty} x_n^{(1)}=+\infty, \quad \lim\limits_{n \rightarrow \infty} x_n^{(2)}=+\infty $$ and $$\lim\limits_{n \rightarrow \infty} \sin{\left(x_n^{(1)}\right)^2} \ne \lim\limits_{n \rightarrow \infty} \sin{\left(x_n^{(2)}\right)^2}$$ what means that $\lim\limits_{x \rightarrow +\infty} \sin{x^{2}}$ does not exist.