Questions: lim as x approaches infinity of (2^(x+1) + 1) / (3 + 2^x)

Solution Steps

To find the limit of the given expression as \( x \) approaches infinity, we can divide the numerator and the denominator by the highest power of 2 present in the expression, which is \( 2^x \). This will simplify the expression and allow us to evaluate the limit as \( x \) approaches infinity.

To find the limit of the expression \(\lim _{x \rightarrow \infty} \frac{2^{x+1}+1}{3+2^{x}}\), we start by simplifying it. We divide both the numerator and the denominator by \(2^x\), the highest power of 2 in the expression:

\[ \frac{2^{x+1} + 1}{3 + 2^x} = \frac{2 \cdot 2^x + 1}{3 + 2^x} = \frac{2 + \frac{1}{2^x}}{\frac{3}{2^x} + 1} \]

As \(x\) approaches infinity, \(\frac{1}{2^x}\) and \(\frac{3}{2^x}\) both approach 0. Therefore, the expression simplifies to:

\[ \lim _{x \rightarrow \infty} \frac{2 + \frac{1}{2^x}}{\frac{3}{2^x} + 1} = \frac{2 + 0}{0 + 1} = 2 \]

Final Answer