Question

find the limits of 1^n and 2^n as n goes to infinity
lim_{n\to\infty}1^n
lim_{n\to\infty}2^n

Explanation:

Step1: Analyze the limit of \(1^n\)

For any real - number \(n\), \(1^n = 1\). So, \(\lim_{n
ightarrow\infty}1^n=\lim_{n
ightarrow\infty}1\).
Since the function \(y = 1\) is a constant function, and for a constant function \(y = C\), \(\lim_{n
ightarrow\infty}C=C\). So, \(\lim_{n
ightarrow\infty}1^n = 1\).

Step2: Analyze the limit of \(2^n\)

We know the formula for the limit of an exponential function \(y = a^n\) (\(a>0\)). When \(a> 1\), \(\lim_{n
ightarrow\infty}a^n=\infty\).
Here \(a = 2>1\), so \(\lim_{n
ightarrow\infty}2^n=\infty\).

Answer:

\(\lim_{n
ightarrow\infty}1^n = 1\), \(\lim_{n
ightarrow\infty}2^n=\infty\)