Convergent limit $\lim_{n\to+\infty} \left(\frac{(3^n-1)^2-9^n}{\sqrt{9^n+n^9}}\right)$ which seems divergent by graph.

Question

Question $$\lim_{n\to+\infty} \left(\frac{(3^n-1)^2-9^n}{\sqrt{9^n+n^9}}\right)$$ My solution $$=\lim_{n\to+\infty} \left(\frac{9^n-2\cdot3^n+1-9^n}{\sqrt{9^n\cdot(1+\frac{n^9}{9^n})}}\right)$$ $$=\lim_{n\to+\infty} \left(\frac{-2+\frac{1}{3^n}}{\sqrt{(1+\frac{n^9}{9^n})}}\right)$$ $$= \frac{0-2}{\sqrt{1+0}}$$ $$=-2$$

But the graph of this sequence looks divergent.

Answer 1

Your work is fine, as an alternative way to check the result we can proceed as follows

• $(3^n-1)^2-9^n=((3^n-1)+3^n)((3^n-1)-3^n)\sim -2\cdot 3^n$
• $\sqrt{9^n+n^9}\sim \sqrt{9^n}=3^n$

that is

$$\frac{(3^n-1)^2-9^n}{\sqrt{9^n+n^9}} \sim \frac{ -2\cdot 3^n}{3^n} \to -2$$

As noticed, the problem with the graph is a numerical issue since exponential terms grow very fast (here a better graph obtained by WA). As a suggestion, we should be very careful when evaluating or check limits by graphs, better to try with alternative ways to obtain the result.