- 176

- 0

**1. Homework Statement**

Compute the limit [tex]\lim_{x \rightarrow 0} (1 + 3x)^{3/x}[/tex].

**2. Homework Equations**

**3. The Attempt at a Solution**

[tex]\lim_{x \rightarrow 0} (1 + 3x)^{3/x} = \lim_{x \rightarrow 0} e^{\frac{3}{x} \ln (1 + 3x)}[/tex]

[tex]\lim_{x \rightarrow 0} \frac{3}{x} \ln (1 + 3x) = 3 \lim_{x \rightarrow 0} \frac{ln (1 + 3x)}{x}[/tex]

[tex]= 3 \lim_{x \rightarrow 0} \frac{\frac{3}{1 + 3x}}{1}[/tex]

[tex]= 3 \lim_{x \rightarrow 0} \frac{3}{1 + 3x}[/tex]

[tex]=9[/tex]

[tex]\lim_{x \rightarrow 0} (1 + 3x)^{3/x} = e^9[/tex]

I just need to know if this is right.

Last edited: