Limits of x* when n and m Approach Infinity

In summary, if $\lim_{m\to\infty} y_{m+1}=x^*$, then we can say that $\lim_{n\to\infty} y_n=x^*$ as well. This is because the limit is independent of the variable used, and we can show that any subsequence must also converge to the same limit.
  • #1
ozkan12
149
0
if $\lim_{{n}\to{\infty}} {y}_{m+1}={x}^{*}$, Can we say that $\lim_{{n}\to{\infty}} {y}_{n-1}={x}^{*}$...Please pay attention, I say m and n...
 
Physics news on Phys.org
  • #2
ozkan12 said:
if $\lim_{{n}\to{\infty}} {y}_{m+1}={x}^{*}$, Can we say that $\lim_{{n}\to{\infty}} {y}_{n-1}={x}^{*}$...Please pay attention, I say m and n...

Did you mean to write

If
$$\lim_{m\to\infty}y_{m+1}=x^*,$$
can we say that
$$\lim_{n\to\infty}y_{n-1}=x^*?$$

Because, if so, the answer is yes, if $n$ and $m$ are just independent integers going to infinity. The variable used in the limit is just like a "dummy variable". You can also show that if $n$ is an arbitrary integer, and
$$\lim_{n\to\infty}y_n=x^*,$$
then any subsequence must also converge:
$$\lim_{m\to\infty}y_{n_m}=x^*.$$
 
  • #3
Dear Ackbach

How if $\lim_{{m}\to{\infty}} {y}_{m+1}={x}^{*}$ we can say that $\lim_{{n}\to{\infty}} {y}_{n}={x}^{*}$ ? I didnt understand...Please can you explain...Thank you for your attention
 
Last edited:

FAQ: Limits of x* when n and m Approach Infinity

What is the definition of a limit?

A limit is the value that a function or sequence approaches as its input or index approaches a certain value. In this case, we are looking at the limit of a function as the inputs approach infinity.

How do you determine the limit of a function?

To determine the limit of a function, we can use various methods such as algebraic manipulation, graphing, or using limit laws. In the case of x* when n and m approach infinity, we can use the limit laws to simplify the expression and determine the limit.

What is the significance of using infinity in limits?

Using infinity in limits allows us to evaluate the behavior of a function as its inputs approach an infinitely large value. This can help us understand the overall trend or behavior of a function as it approaches infinity.

Can a function have a limit when n and m approach infinity?

Yes, a function can have a limit when n and m approach infinity. This is known as an infinite limit and it means that the function's output is approaching an infinitely large value as its inputs approach infinity.

How does the concept of limits relate to calculus?

Limits are a fundamental concept in calculus and are used to define derivatives and integrals. They help us understand the behavior of a function as it approaches a certain value, which is crucial in solving problems in calculus such as finding the slope of a curve or the area under a curve.

Similar threads

Replies
4
Views
2K
Replies
11
Views
1K
Replies
11
Views
789
Replies
7
Views
2K
Replies
6
Views
2K
Replies
2
Views
664
Replies
9
Views
1K
Back
Top