1. lim 4^n+2^n/4^n+1 +2^n-1 2. lim 3^n+1 +4^n-2/4^n+4 +5^n+1

Đáp án:

$1)$ $\lim \dfrac{4^n+2^n}{4^{n+1}\:+2^{n-1}}$ $=\dfrac{1}{4}$

$2)$ $\lim \dfrac{3^n+1\:+4^n-2}{4^n+4\:+5^n+1}$ $=0$

Giải thích các bước giải:

$1)$ $\lim \dfrac{4^n+2^n}{4^{n+1}\:+2^{n-1}}$

$=\lim \:\dfrac{\dfrac{1}{4}+\dfrac{2^n}{4^{n+1}}}{1+\dfrac{2^{n-1}}{4^{n+1}}}$

$=\dfrac{\lim \dfrac{1}{4}+\dfrac{2^n}{4^{n+1}}}{\lim \:1+\dfrac{2^{n-1}}{4^{n+1}}}$

$=\dfrac{\dfrac{1}{4}}{1}$

$=\dfrac{1}{4}$

$2)$ $\lim \dfrac{3^n+1\:+4^n-2}{4^n+4\:+5^n+1}$

$=\lim \:\dfrac{\left(\dfrac{3}{5}\right)^n+\dfrac{1}{5^n}+\left(\dfrac{4}{5}\right)^n-\dfrac{2}{5^n}}{\left(\dfrac{4}{5}\right)^n+\dfrac{4}{5^n}+1+\dfrac{1}{5^n}}$

$=\dfrac{\lim \left(\dfrac{3}{5}\right)^n+\dfrac{1}{5^n}+\left(\dfrac{4}{5}\right)^n-\dfrac{2}{5^n}}{\lim \left(\dfrac{4}{5}\right)^n+\dfrac{4}{5^n}+1+\dfrac{1}{5^n}}$

$=\dfrac{0}{1}$

$=0$

Đáp án:

$1) \dfrac{1}{4}\\ 2) 0.$

Giải thích các bước giải:

$1) \lim \dfrac{4^n+2^n}{4^{n+1}+2^{n-1}}\\ =\lim \dfrac{4^n+2^n}{4.4^n+2^{-1}.2^n}\\ =\lim \dfrac{4^n+2^n}{4.4^n+\dfrac{1}{2}.2^n}\\ =\lim \dfrac{1+\left(\dfrac{1}{2}\right)^n}{4+\dfrac{1}{2}.\left(\dfrac{1}{2}\right)^n}\\ =\dfrac{1}{4}\\ 2) \lim \dfrac{3^{n+1}+4^{n-2}}{4^{n+4}+5^{n+1}}\\ =\lim \dfrac{3.3^n+\dfrac{1}{16}.4^n}{4^4.4^n+5.5^n}\\ =\lim \dfrac{3.\left(\dfrac{3}{5}\right)^n+\dfrac{1}{16}.\left(\dfrac{4}{5}\right)^n}{4^4.\left(\dfrac{4}{5}\right)^n+5}\\ =\dfrac{0}{5}\\ =0.$