Proving (1+x)^n approaches 1 + nx when x goes to zero

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In summary, the conversation is about proving the statement that for any real number n, the limit of (1+x)^n as x approaches 0 is 1. The person asking the question is unsure of how to prove it and suggests using binomial expansion or the concept of limits. The other person suggests using the binomial theorem and discussing the meaning of the statement before attempting to prove it. They also suggest two possible rigorous versions of the statement to prove.
  • #1
U.Renko
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Homework Statement


I've heard this affirmation in an gravitation lecture. I thought it was interesting and decided to check it out. It turns out to be true so I decided to prove it.

I'm not a mathematician so proving stuff is not my department. Still I try to do some proofs just for fun. I hope I don't make your eyes bleed or something.


Homework Equations


[itex]\lim_{x \rightarrow 0} (1 + x)^n[/itex]


The Attempt at a Solution


I'm not sure if I should try prooving by epsilon-delta or by mathematical induction (if that is even possible)
I did try by mathematical induction but it didnt look convincing

How would you do it?
 
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  • #2
[itex]\lim_{x \to 0} (1 + x)^n [/itex]

Apply binomial expansion to this equation and then the concept of limits.
 
  • #3
Try Binomial Theorem.

While it is correct to say [itex]{(1+x)}^n \approx 1+nx[/itex] for sufficiently small x, the actual limit of the expression as x tends to zero is simply 1.
 
  • #4
It's impossible to prove this until you rigorized what the statement means.

I think the best rigorous version of this statement is that y=1+nx is the tangent line of y=(1+x)n at the point x=0.

Or, it could mean

[tex]\lim_{x\rightarrow 0} \frac{(1+x)^n-(1+nx)}{x}=0[/tex]

which means that the functions are equal up to first order.

Try to prove these two statements.
 

FAQ: Proving (1+x)^n approaches 1 + nx when x goes to zero

What does it mean for (1+x)^n to approach 1 + nx when x goes to zero?

When x approaches zero, it means that the value of x is getting closer and closer to zero, but not actually reaching zero. In this case, when we say (1+x)^n approaches 1 + nx, it means that as x gets closer to zero, the expression (1+x)^n will have a value that is very close to the expression 1 + nx, but not exactly equal to it.

Why is it important to prove that (1+x)^n approaches 1 + nx when x goes to zero?

This proof is important because it helps us understand the behavior of the expression (1+x)^n as x approaches zero. This is especially useful in mathematical and scientific calculations where we need to make approximations or estimations.

What is the mathematical process for proving that (1+x)^n approaches 1 + nx when x goes to zero?

The proof involves using the binomial theorem to expand the expression (1+x)^n and then rearranging the terms to show that it is equal to 1 + nx. This can be done using algebraic manipulations and the properties of limits.

Can you provide an example of how to prove that (1+x)^n approaches 1 + nx when x goes to zero?

Yes, for example, if we take the limit as x approaches zero of the expression (1+x)^2, we can use the binomial theorem to expand it to 1 + 2x + x^2. Then, as x approaches zero, the value of x^2 becomes negligible and the expression simplifies to 1 + 2x, which is the same as 1 + nx when n=2.

What are the real-life applications of understanding and proving that (1+x)^n approaches 1 + nx when x goes to zero?

This concept is used in various fields such as physics, engineering, and economics. For example, in physics, it can be used to approximate the trajectory of a projectile or the behavior of a spring. In economics, it can be used to estimate the growth of investments or the effects of inflation. Understanding this concept also helps in understanding the behavior of functions and making predictions based on their limits.

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