Finding the limit of a function with e and ln

In summary, e and ln are closely related as the inverse function of each other and are used to simplify and evaluate complex expressions in finding the limit of a function. To find the limit using these tools, we can rewrite the expression using the properties of logarithms and then use algebraic manipulation and substitution. Although e and ln are powerful tools, they may not always work for all types of functions and may not result in a simplified expression. However, they have practical applications in various fields such as physics, engineering, and economics.
  • #1
trap
53
0
lim as x->infinity [e^x + x] ^(1/x)

Can anyone help me on this please, thanks.
 
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  • #2
The limit is "e"...To convince yourself,use the fact that the limit & the natural logarithm commute.

Daniel.
 
  • #3
can you take me one step further than that, to get the answer e?
 
  • #4
i got it now, thanks for the help
 
  • #5
There would be another way to do it,directly,without use of [itex] ln [/itex].Just factor e^{x} and then use the definition of "e":
[tex] \lim_{u\rightarrow +\infty}(1+\frac{1}{u})^{u}=e [/tex]

Daniel.
 

1. What is the relationship between e and ln in finding the limit of a function?

The natural logarithm (ln) is the inverse function of the exponential function (e). This means that e raised to the power of ln x will always give us back x. In finding the limit of a function, e and ln are used to simplify and evaluate complex expressions.

2. How do you use e and ln to find the limit of a function?

To find the limit of a function using e and ln, we can use the properties of logarithms to rewrite the expression in a simpler form. This typically involves using the natural logarithm to remove exponents and simplify the expression. Then, we can use algebraic manipulation and substitution to evaluate the limit.

3. What is the difference between e and ln in finding the limit of a function?

E is a mathematical constant with a value of approximately 2.71828, while ln is a mathematical function used to find the natural logarithm of a number. In finding the limit of a function, e is used as the base of the exponential function, while ln is used to simplify expressions and evaluate the limit.

4. Are there any limitations to using e and ln in finding the limit of a function?

While e and ln are powerful tools for evaluating limits, they may not work for all types of functions. Some limits may require alternative methods, such as L'Hôpital's rule or graphing, to find the limit. Additionally, using e and ln may not always result in a simplified expression, making it difficult to evaluate the limit.

5. What are some real-world applications of finding the limit of a function using e and ln?

Finding the limit of a function using e and ln is commonly used in calculus and other advanced mathematics, but it also has practical applications in fields such as physics, engineering, and economics. For example, it can be used to model growth and decay in population, interest rates, and radioactive decay.

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