Evaluate the Limit limit as x approaches 0 of (1+3x)^(1/x)

Problem

(lim_x→0)(1+3*x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which yields 1

2. Rewrite the expression using the natural logarithm and the exponential function to move the variable out of the exponent.

y=(1+3*x)1/x

y=eln((1+3*x)1/x)

3. Apply the power rule for logarithms to simplify the exponent.

y=e(1/x*ln(1+3*x))

4. Evaluate the limit of the exponent as x→0 using L'Hôpital's Rule, since it results in the indeterminate form 0/0

(lim_x→0)(ln(1+3*x)/x)

5. Differentiate the numerator and the denominator with respect to x

d(ln(1+3*x))/d(x)=3/(1+3*x)

d(x)/d(x)=1

6. Calculate the limit of the resulting fraction.

(lim_x→0)(3/(1+3*x))=3

7. Substitute the result back into the exponential form.

(lim_x→0)(eln(1+3*x)/x)=e3

Final Answer

(lim_x→0)(1+3*x)=e3

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