Proving the Limit of f(x) as x->0 and Lim[e^n-(1+1/n)^n^2]

  • Thread starter azatkgz
  • Start date
  • Tags
    Limit
In summary, the conversation discusses proving that the function f(x) has no limit as x tends to 0 using mathematical formulas and the definition of limit. The use of epsilon and delta neighborhoods is also mentioned, as well as the characterization of limits in terms of sequences. The conversation ends with a request for a solution and a reminder to try different approaches.
  • #1
azatkgz
186
0
f(x)=1,xeQ
f(x)=0,x in not eQ

It's easy to understand that this function has no limit as x tends to 0,but how we can prove it with mathematical formulas.

And what's the lim[e^n-(1+1/n)^n^2]
 
Last edited:
Physics news on Phys.org
  • #2
Prove it with the definition of limit.

Can you find an epsilon such that no delta-nbhd of 0 is mapped entirely in an epsilon-nbhd of f(0)=1?

Or use the characterisation in terms of sequences. The limit is f(0) iff for every sequence converginf to 0, the image sequence converges to f(0). Can you find a sequence such that the image sequence does not converge to f(0)?
 
Last edited:
  • #3
quasar987,
I could not solve with sequences.Can u please post the solution.
 
  • #4
So far you haven't shown that you have tried anything at all! What have you tried?

Have you thought about what f(x) is if x is close to 0 and rational?
What f(x) is if x is close to 0 and irrational?
COULD f(x) be "close" to some limit if x is any number close to 0?
 

FAQ: Proving the Limit of f(x) as x->0 and Lim[e^n-(1+1/n)^n^2]

1. What is the definition of a limit in calculus?

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value or point. In other words, it is the value that a function approaches as the input gets closer and closer to a specific value or point. In calculus, limits are used to determine the behavior of a function near a specific point and are an essential tool for understanding the behavior of functions.

2. How do you prove the limit of a function as x approaches 0?

To prove the limit of a function as x approaches 0, we use the definition of a limit. This involves showing that for any given value of epsilon (ε), we can find a corresponding value of delta (δ) such that if the distance between x and 0 is less than delta, then the distance between f(x) and the limit is less than epsilon. In other words, we need to show that the function values approach a specific value as the input gets closer to 0.

3. What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the behavior of a function as the input approaches the specific value from one direction (either from the left or the right). On the other hand, a two-sided limit takes into account the behavior of the function as the input approaches the specific value from both the left and right sides. In some cases, the one-sided limit and two-sided limit may be equal, but in others, they may be different.

4. How do you prove the limit of a complicated function?

To prove the limit of a complicated function, we can use algebraic manipulation, trigonometric identities, or other calculus techniques such as L'Hôpital's rule. We can also use the properties of limits, such as the sum, product, and quotient rules, to simplify the function and evaluate the limit. It is important to carefully follow the steps of the proof and justify each step using the definition of a limit.

5. How do you prove the limit of an exponential function?

To prove the limit of an exponential function, we can use the properties of limits, specifically the limit of a composite function. By writing the exponential function as a composite function, we can use the known limit of the inner function to evaluate the limit of the entire function. For example, in the given function, we can write e^n as the exponential function e^x, and (1+1/n)^n^2 as the composite function (1+1/n)^x. Then, we can use the known limit of e^x as x approaches infinity to evaluate the limit of the given function as n approaches infinity.

Similar threads

I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
Replies
9
Views
2K
I What is the difference between these two power series theorems?
Replies
5
Views
732
I Proof of 1/(x^2) Not Having Limit at 0
Replies
5
Views
2K
I Multivariable fundamental calculus theorem in Wald
Replies
2
Views
2K
MHB Proving the Squeeze Theorem - Finding Limits Using Squeeze Theorem
Replies
6
Views
1K
B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
Replies
19
Views
3K
I Express the limit as a definite integral
Replies
16
Views
3K
B Why do I not understand what seems to be basic Calculus?
Replies
11
Views
2K
B Naturals or Reals When Taking Limits to Obtain the Value of Euler's Number e?
Replies
11
Views
2K
I How does the ratio test fail and the root test succeed here?
Replies
6
Views
2K

Hot Threads

Recent Insights

Back
Top