Proving Continuity of exp(x) at c=0

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In summary, to prove that e^x is continuous at c = 0 using the delta-epsilon method, we can use the conditions (a) and (b) given in the problem. By setting y=e in (a), we can find a delta that is a function of the N(epsilon) of that limit. Then, using condition (b), we can show that for any given epsilon, there exists a delta such that |exp(x) - exp(0)| < epsilon, proving the continuity of e^x at c = 0.
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skoomafiend
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Homework Statement


use delta, epsilon to prove that e^x is continuous at c = 0


Homework Equations



(a) for y>0, lim_n-> inf, y^(1/n) = 1
(b) for x < y, exp(x) < exp(y)

The Attempt at a Solution



im not sure how to approach this problem.
i have,
|exp(x) - exp(0)|= |exp(x) - 1|
so then exp(x) < 1 + ε
for δ > 0,
exp(δ) < 1 + ε

so then, i would set δ=ln(1+ε) for the proof?

also I am not sure how to use relevant eqn (a) to help with the problem. some insight would be appreciated. thank you.
 
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  • #2
You are not supposed to use log; you are supposed to use the two conditions (a) and (b); first you realize that the limit in (a) is fairly close to your limit, if you set y=e, then you can use that limit to come up with a delta which is a function of the N(epsilon) of that limit.
 

FAQ: Proving Continuity of exp(x) at c=0

What does it mean for a function to be continuous at a specific point?

Continuity at a point means that the value of the function at that point is equal to the limit of the function as the independent variable approaches that point. In other words, there are no sudden jumps or breaks in the function at that point.

How is continuity of a function at a point proven mathematically?

To prove continuity at a point, we need to show that the limit of the function as the independent variable approaches that point exists and is equal to the value of the function at that point. This can be done using the epsilon-delta definition of continuity.

What is the epsilon-delta definition of continuity?

The epsilon-delta definition of continuity states that for a function f(x) to be continuous at a point c, for any given epsilon (ε) greater than 0, there exists a corresponding delta (δ) greater than 0 such that if the absolute value of (x-c) is less than delta, then the absolute value of (f(x)-f(c)) is less than epsilon.

How do we prove continuity of exp(x) at c=0?

To prove continuity of exp(x) at c=0, we need to show that the limit of exp(x) as x approaches 0 exists and is equal to the value of exp(0), which is 1. This can be done using the epsilon-delta definition of continuity, by choosing an appropriate delta value that satisfies the definition.

Why is proving continuity at c=0 important for the function exp(x)?

Proving continuity at c=0 for exp(x) is important because it is the base point for the entire exponential function. Without continuity at this point, the function would not be well-defined and would not follow the expected behavior. It is also important for the continuity of the natural logarithm function, which is the inverse of the exponential function.

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