Question

Find the limit. $$ \lim _{x \rightarrow 1}\left(\ln 3 x+e^{x}\right) $$

Short answer

The limit of the function as \(x\) approaches 1 is approximately 3.8166.

Step-by-step solution

Recognizing the Problem Type

This is a limit problem where \(x\) is approaching 1. The function is \(\ln 3x + e^x\).

Substituting x

Substitute \(x = 1\) into the function: \(\ln 3(1) + e^1 = \ln 3 + e\).

Evaluating the Natural Logarithm and Exponential

Evaluate \(\ln 3\) and \(e\), where \(e\) is Euler's Number (approx. 2.718). That gives us \(1.0986 + 2.718\).

Adding the Results

Adding these two numbers gives the final result: \(1.0986 + 2.718 = 3.8166\).