6.120 . lim _(xarrow 0)((1+3x)/(1+x))^(5)/(x)

To evaluate the limit \(\lim_{x \to 0} \left( \frac{1 + 3x}{1 + x} \right)^{\frac{5}{x}}\), we can use the property of limits involving exponentials and logarithms. Let's denote the expression inside the limit as : Taking the natural logarithm of both sides, we get: Using the property of logarithms, we can bring the exponent in front: Now, we need to evaluate the limit of the right-hand side as approaches 0: We can simplify the argument of the logarithm: As approaches 0, approaches 0, so: Thus, the expression becomes: Therefore, the natural logarithm of the limit is 10: Exponentiating both sides to solve for : So, the limit is: