# Exploring the Continuity Property of e in Limits

• brunokabahizi
In summary, the conversation discusses the property of continuity that allows the fact that if the limit of a function is equal to a constant, then the limit of the exponential function with the same exponent will also be equal to that constant. This property is defined as a function being continuous at a certain point, where the function must satisfy the conditions of existence and equality of its limit at that point. The conversation also mentions a theorem in Calculus that discusses the composition of functions, stating that the limit of the composition of two functions is not always equal to the limit of the individual functions, unless the function is continuous at the point of composition. Therefore, continuity is the key factor in this fact.
brunokabahizi

## Homework Statement

If the limx→b f(x)=c, then limx→b ex= ec. What property of the function g(x)=ex allows this fact?

## The Attempt at a Solution

Is it just because e is a constant?

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No. e is a constant, but e^x is not. and probably you meant $lim_{x→b}e^{f(x)}=e^c$. well, I think the answer is continuity.

Yes, you're right. I did mean limx→b ef(x)= ec. But how does continuity allow it? Excuse me for being incompetent.

Did you consider looking up "continuous" in your text?

That property pretty much is the definition of "continuous":

The function f(x) is continuous at x= a if and only if
(1) f(a) exists.
(2) $\lim_{x\to a} f(x)$ exists.
(3) $\lim_{x\to a} f(x)= f(a)$.

Since (3) pretty much implies the left and right sides exist, of only that is given as the definition- but that's really "shorthand".

well, I meant that the function needs to be continuous at the point x=c. There's a theorem in Calculus that talks about the composition of functions. the theorem states that if $lim_{x→b}f(x)=c$ and $lim_{x→a}g(x)=b$ then $lim_{x→a} fog(x)=c$ is NOT true in general, but this law holds if f is continuous at x=b.
That's why I said continuity is the key. The theorem seems to be easy to be proved though.

## What is the constant e?

The constant e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828. It is a fundamental constant in mathematics and has many important applications in calculus, exponential functions, and complex analysis.

## Where does the constant e come from?

The constant e was first introduced by the Swiss mathematician Leonhard Euler in the 18th century. It arises naturally in the study of compound interest and continuously compounded growth.

## What are the properties of the constant e?

The constant e has several important properties, including being the base of the natural logarithm function, being the unique number with the property that its derivative is equal to itself, and being the limit of (1 + 1/n)^n as n approaches infinity.

## How is the constant e used in mathematics?

The constant e is used extensively in mathematics, particularly in calculus and complex analysis. It is used to model continuous growth and decay, and to simplify calculations involving exponential and logarithmic functions.

## What are some real-life applications of the constant e?

The constant e has many real-life applications, including in finance, biology, physics, and engineering. It is used to model population growth, radioactive decay, and the spread of infectious diseases. It is also used in the calculation of interest and compound interest in financial transactions.

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