Math Problem Statement

evaluate

Evaluate the limit: lim(x → 0) (a^x + x)^(1/x)

Solution

To evaluate the limit:

limx0(ax+x)1/x\lim_{x \to 0} \left( a^x + x \right)^{1/x}

we can start by taking the natural logarithm to simplify the expression. Let:

L=limx0(ax+x)1/xL = \lim_{x \to 0} \left( a^x + x \right)^{1/x}

Taking the natural logarithm of both sides, we get:

lnL=limx0ln(ax+x)x\ln L = \lim_{x \to 0} \frac{\ln(a^x + x)}{x}

We can apply L'Hôpital's rule to evaluate this limit because it is of the indeterminate form 00\frac{0}{0} as x0x \to 0.

Let's find the derivative of the numerator and the denominator. The derivative of ln(ax+x)\ln(a^x + x) with respect to xx is:

ddxln(ax+x)=axlna+1ax+x\frac{d}{dx} \ln(a^x + x) = \frac{a^x \ln a + 1}{a^x + x}

The derivative of the denominator xx with respect to xx is 1.

Now applying L'Hôpital's rule:

lnL=limx0axlna+1ax+x\ln L = \lim_{x \to 0} \frac{a^x \ln a + 1}{a^x + x}

As x0x \to 0, ax1a^x \to 1, so we have:

lnL=1lna+11+0=lna+1\ln L = \frac{1 \cdot \ln a + 1}{1 + 0} = \ln a + 1

Therefore,

L=elna+1=aeL = e^{\ln a + 1} = a \cdot e

So the limit is:

limx0(ax+x)1/x=ae\lim_{x \to 0} \left( a^x + x \right)^{1/x} = a \cdot e

Would you like more details or have any questions? Here are some related questions to consider:

  1. What is L'Hôpital's rule and how is it used in limit evaluation?
  2. How does taking the natural logarithm help in simplifying limits involving exponents?
  3. Can you provide another example of a limit that requires L'Hôpital's rule for evaluation?
  4. What are indeterminate forms and why are they important in calculus?
  5. How does the behavior of axa^x change as x0x \to 0 for different values of aa?

Tip: When evaluating complex limits, taking the natural logarithm can often simplify the problem by turning exponentiation into multiplication, which can then be managed using L'Hôpital's rule or other techniques.

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Exponential Functions
L'Hopital's Rule

Formulas

lim(x → 0) (a^x + x)^(1/x)
L'Hopital's Rule: lim(x → c) f(x)/g(x) = lim(x → c) f'(x)/g'(x)

Theorems

L'Hopital's Rule
Properties of Exponential Functions
Logarithmic Differentiation

Suitable Grade Level

Grades 11-12, College Calculus