Limit Question Solution | Simplifying Infinite Limit with e^2 Result

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In summary, a limit in mathematics is the value that a function or sequence approaches as the input or variable approaches a specific value. To solve a limit question with an e^2 result, one can use the rules of limits and the properties of exponentials. There are various techniques for simplifying infinite limits, including L'Hopital's rule, factoring, and rationalizing. The exponential function e^2 plays a crucial role in limit questions due to its derivative of itself and its value as an irrational number. Solving limit questions with e^2 result has practical applications in fields such as physics, engineering, and economics, where it is used to model exponential growth and decay and solve optimization problems.
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talolard
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Homework Statement


I've done thsi limit, i think I'm half right. I'm sure I have done something wrong, but I'm not sure what.

[tex] lim_{x-> \infty} ( \frac {x+lnx}{x-lnx})^(x/lnx) [/tex]


The Attempt at a Solution


[tex] lim_{x-> \infty} ( \frac {x+lnx}{x-lnx})^{x/lnx} = lim_{x-> \infty} ( \frac {x(1 +\frac {lnx}{x})}{x(1- \frac {lnx}{x})})^{x/lnx}= [/tex]
[tex]
t= \frac {x}{lnx}; lim_{t-> \infty} (\frac {1 + \frac {1}{t}} {1 - \frac{1}{t}})^t= [/tex]
[tex] \frac {e}{e^{-1}}=e^2 [/tex]
 
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That's right.
 

1. What is a limit in mathematics?

A limit in mathematics is the value that a function or sequence approaches as the input or variable approaches a specific value. It is a fundamental concept in calculus and is used to describe the behavior of functions near points where they are not defined or are undefined.

2. How do you solve a limit question with e^2 result?

To solve a limit question with an e^2 result, you can use the rules of limits and the properties of exponentials. First, rewrite the function using the natural logarithm, ln(x), to simplify the expression. Then, use the properties of logarithms to manipulate the expression into a form that can be evaluated using the rules of limits. Finally, take the limit of the simplified expression to find the solution.

3. What are the common techniques for simplifying infinite limits?

Some common techniques for simplifying infinite limits include using the L'Hopital's rule, factoring, and rationalizing the numerator or denominator. These techniques can help to manipulate the expression into a form that can be evaluated using the rules of limits.

4. Can you explain the role of e^2 in limit questions?

e^2, or the exponential function with base e, is often used in limit questions because it is a special function that has a derivative of itself. This makes it useful for solving limits involving exponential functions. Additionally, e^2 is an irrational number with a value of approximately 7.389, which makes it a convenient choice for simplifying expressions involving other irrational numbers.

5. What are the applications of solving limit questions with e^2 result in real life?

Solving limit questions with e^2 result has various applications in real life, particularly in the fields of physics, engineering, and economics. It can be used to model exponential growth and decay, such as in population growth or radioactive decay. It is also used in the study of rates of change and optimization problems, which are important in many real-world scenarios.

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