What is the Limit of (1+3x)^(1/x) as x Approaches 0+?

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In summary, the conversation discusses the limit \lim_{x\rightarrow 0^+}(1+3x)^{1/x} and the process of making substitutions to simplify it. The concept of Euler's number and the definition of the limit are also mentioned. A link is provided for further understanding of the topic."
  • #1
jdstokes
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[itex]\lim_{x\rightarrow 0^+}(1+3x)^{1/x}[/itex]

Thanks.
 
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  • #2
hello there

firstly i would make a substitution such as
[itex]x=\frac{1}{u}[/itex]
[itex]\lim_{x\rightarrow 0^+}(1+3x)^{1/x}=\lim_{u\rightarrow\infty}(1+\frac{3}{u})^{u}[/itex] you do know what this limit is equal to right?

steven
 
  • #3
It might be better to make the substitution [tex]x= \frac{1}{3u}[/tex] so that [tex]1+ 3x= 1+ \frac{1}{u}[/tex] and [tex]\frac{1}{x}= 3u[/tex]. Then the limit becomes
[tex]\lim_{x\rightarrow 0^+}(1+3x)^{1/x}=\lim_{u\rightarrow\infty}(1+\frac{1}{u})^{3u}= \{\lim_{u\rightarrow\infty}(1+ \frac{1}{u})^u\}^3[/tex].
 
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  • #4
I still don't know where to go with this. I seem to keep getting a [itex](1+ 0)^\infty[/itex] situation.
 
  • #5
Do you know the definition of "e" (Euler's number)...?

Daniel.
 
  • #6

What is a limit in mathematics?

A limit in mathematics is the value that a function or sequence approaches as the input or index approaches a certain value. It is used to study the behavior of a function or sequence near a specific point and is an important concept in calculus and analysis.

Why do we need hints for limits?

Limits can be challenging to solve, especially for complex functions. Hints can provide guidance and help students approach the problem in a structured manner. They can also help identify errors in the solution process.

How can I find a hint for a limit?

There are many resources available to help with solving limits, such as textbooks, online tutorials, and study groups. You can also ask your teacher or peers for hints or guidance.

What are some common techniques for solving limits?

Some common techniques for solving limits include direct substitution, factoring, rationalization, and using L'Hopital's rule. It is important to choose the right technique depending on the type of function and the limit at hand.

Why is understanding limits important?

Limits are used in many areas of mathematics, such as calculus, differential equations, and analysis. They are also used in other fields such as physics and engineering. Understanding limits is crucial for solving complex problems and for furthering our understanding of mathematical concepts.

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