Compute the limit : $\lim\limits_{n \to \infty} \dfrac{5 + (-1)^n}{ \sqrt{n} +7}$
My work:
Since $(-1)^n$ can be either -1 or +1 so, with Sandwich Law the numerator can be within 4 and 6 i.e $$ 4 \leqslant 5 + (-1)^n \leqslant 6$$. The denominator grows accordingly with $\sqrt{n}$. Using this info how can I proceed further ?
Best Answer
My approach. Being
$$\lim_{n \to \infty} \frac{5 + (-1)^n}{\sqrt{n}+7}=\color{magenta}{\lim_{n \to \infty} \frac{5}{\sqrt{n} +7}}+\color{blue}{\lim_{n \to \infty} \frac{(-1)^n}{\sqrt{n} +7}}$$ The sequence $(-1)^n$ is bounded. The product of a finite and an infinitesimal sequence gives a null limit. Hence the blue limit is $0$. The sequence is easy to solve because $$\color{magenta}{\lim_{n \to \infty} \frac{5}{\sqrt{n} +7}}\equiv 0$$ Definitively:
$$\lim_{n \to \infty} \frac{5 + (-1)^n}{\sqrt{n}+7}=0$$