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spektah
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Homework Statement
Some on ehelp me prove this in detailed format using the knowledge of limits.
Homework Equations
lim(x-->infinity)(1+1/x)^x=e
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malawi_glenn said:So you can "only" show that this limit DOES exists.
dynamicsolo said:You can show that the limit is e, but I think you need something at least as strong as L'Hopital's Rule (after taking the natural logarithm of the expression and arranging the result into appropriate form) to prove it. I'm not aware of a nice shortcut.
malawi_glenn said:According to my books, this is the definition of e, they (and me) could be wrong. I mean, the natural logarithm requires that you already have e and e^x right?
A calculus limit proof is a mathematical method used to show the behavior of a function as it approaches a certain point or value. It involves evaluating the limit of the function and providing a rigorous mathematical argument to prove that the limit exists and has a specific value.
The first step in approaching a calculus limit proof is to understand the definition of a limit and the properties of limits. Then, you can use algebraic manipulations, trigonometric identities, and other techniques to simplify the expression and evaluate the limit. Finally, you need to provide a logical and mathematical argument to prove the limit exists and has a specific value.
A limit and a derivative are both concepts in calculus, but they have different meanings and applications. A limit is used to describe the behavior of a function as it approaches a certain point, while a derivative is used to describe the instantaneous rate of change of a function at a specific point. In other words, a limit looks at the overall behavior of a function, while a derivative looks at the local behavior.
Proving a calculus limit is essential in understanding the behavior of a function and its properties. It allows us to make accurate predictions about the function and its values, which is crucial in many areas of mathematics, science, and engineering. Additionally, proving a limit is a fundamental skill in calculus and is often required in advanced courses.
The epsilon-delta definition of a limit is a rigorous way to define the limit of a function. It states that for a given function f(x) and a point c, the limit of f(x) as x approaches c is L if for every positive number epsilon, there exists a positive number delta such that if the distance between x and c is less than delta, then the distance between f(x) and L is less than epsilon. In simpler terms, it means that as x gets closer and closer to c, f(x) gets closer and closer to L.