Limit with x in both base and exponenet

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In summary, the conversation is about solving a limit involving a constant and using L'Hôpital's Rule to prove that the limit is equal to e^(a+1).
  • #1
Yankel
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Dear all,

I am trying to solve the following limit:

\[\lim_{x\rightarrow 0}(e^{ax}+x)^{\frac{1}{x}}\]

where \[a\] is a constant.

I know that the limit is equal to \[e^{a+1}\] but not sure how to prove it.

Thank you.
 
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  • #2
I would first write:

\(\displaystyle L=\lim_{x\to0}\left(\left(e^{ax}+x\right)^{\frac{1}{x}}\right)\)

Next, take the natural log of both sides, and simplify to get:

\(\displaystyle \ln(L)=\lim_{x\to0}\left(\frac{\ln\left(e^{ax}+x\right)}{x}\right)\)

Now you have the indeterminate form 0/0 and can apply L'Hôpital's Rule. The result you seek will follow.
 

FAQ: Limit with x in both base and exponenet

What is a limit with x in both base and exponent?

A limit with x in both base and exponent is a mathematical concept that represents the value that a function approaches as the independent variable (x) approaches a certain value. In this case, both the base and exponent of the function contain the variable x.

How do you solve a limit with x in both base and exponent?

To solve a limit with x in both base and exponent, you can use algebraic manipulation, substitution, or L'Hopital's rule. First, try simplifying the expression by factoring or canceling common terms. If that doesn't work, you can substitute values for x that approach the limit value. If the limit is indeterminate, you can use L'Hopital's rule to take the derivative of the function and evaluate the limit again.

What are some common examples of limits with x in both base and exponent?

Some common examples of limits with x in both base and exponent include exponential functions, logarithmic functions, and rational functions. For example, the limit of x^x as x approaches 0 is 1, and the limit of ln(x)/x as x approaches infinity is 0.

Why are limits with x in both base and exponent important in mathematics?

Limits with x in both base and exponent are important in mathematics because they help us understand the behavior of functions as the independent variable approaches a certain value. They are also used in calculus to calculate derivatives and integrals, and in other areas of mathematics such as differential equations and complex analysis.

How can limits with x in both base and exponent be applied in real life?

Limits with x in both base and exponent can be applied in real life in various fields such as physics, economics, and engineering. For example, in physics, limits can be used to calculate the velocity and acceleration of an object as it approaches a certain point. In economics, limits can be used to analyze the growth of a population or the value of a stock over time. In engineering, limits can be used to design and optimize structures or systems.

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