Question

Calculate the following limits.
$\underset{n\to \infty }{lim}\frac{5^{n+1}+3^n-2^{2n}}{5^n+2^n+3^{n+2}},nϵN.$

Answer 1

$\underset{n\to \infty }{lim}\frac{5^{n+1}+3^n-2^{2n}}{5^n+2^n+3^{n+2}}$

Dividing both numerator and denominator by $5^n$:-

$=\underset{n\to \infty }{lim}\frac{5+\frac{3^n}{5^n}-\frac{4^n}{5^n}}{1+\frac{2^n}{5^n}+9\frac{3^n}{5^n}}$

Since $\frac{2}{5}<\frac{3}{5}<1$

$⟹\underset{n\to \infty }{lim}\frac{2^n}{5^n}=\underset{n\to \infty }{lim}\frac{3^n}{5^n}=0$

Hence-

$=\underset{n\to \infty }{lim}\frac{5+\frac{3^n}{5^n}-\frac{4^n}{5^n}}{1+\frac{2^n}{5^n}+9\frac{3^n}{5^n}}$

$=\frac{5+0-0}{1+0+0}$

$=5$