# What Is the Limit of cos(1/(10-t)) as t Approaches 10?

• Jan Hill
In summary, the limit of cos(1/10-t) as t approaches 10 does not exist due to oscillation between 1 and -1. The limit of (t^2-100)/t times cos(1/(10-t)) + 100 as t approaches 10 is 100. Using LaTeX can help express mathematical information accurately in text. The limit can also be rewritten in terms of u as t approaches 0. Links are provided for learning how to use LaTeX and advanced options for writing exponents and subscripts.
Jan Hill

## Homework Statement

limit as t-->10 of cos (1/10-t)

## The Attempt at a Solution

Obviously we can't substitute in t = 10 or it would be undefined, so how do we do this. Is there an inequalitly that I haven't considered?

supposing that the function is: $$cos(\frac{1}{10-t})$$
well, the limit does not exist because the cosine function will oscillate crazily between 1 and -1 as t->10.

Last edited:
Thanks. I was looking at

lim t-->10 of (t^2-100)/t times cos1/(10-t) + 100

So since the lim as t-->10 of cos1/(10 - t) is undefined, we're left with 100

By the way, I'd use the proper way of expressing math information with symbols etc. if I knew how to use it but I don't know where to get the symbols etc. to be able to use them. Do you know how to remedy this issue?

Yes that's right, but with a little change of words - it's not because it's "undefined" but because the limit doesn't exist and it's because of this oscillation between -1 and 1 that we can conclude it is finite as the limit approaches, thus the limit is 0 due to the other factor.

It's called Latex and the way you use it is by using the tags [ tex ] ...code... [ /tex ] (without the spaces in the tex tags). You can find some code to use by searching through the Sum symbol in the full text editor when you type out your message.

thank you very much

Do I need to download Latex to my pc? I have looked at it and I see how great it will be but how do I get to use it!?

No you just type it into your message. To get an idea of how to use it, click on the latex I'm going to do to show you how you would do a fraction.

$$\frac{1}{2}$$

Is there any way that we would be able to simplify any parts of the equation and then take the limit? Or its just a substitution problem?

Not really, but if you prefer a limit going to zero rather than 10, let u=t-10

Then you have,

$$\lim_{u\to 0}\frac{u(u+20)}{u+10}cos\left(\frac{-1}{u}\right)+100$$

Jan Hill said:
By the way, I'd use the proper way of expressing math information with symbols etc. if I knew how to use it but I don't know where to get the symbols etc. to be able to use them. Do you know how to remedy this issue?

Here are some links to sites that show how to use LaTeX.
http://heather.cs.ucdavis.edu/~matloff/LaTeX/LookHereFirst.html
http://andy-roberts.net/misc/latex/latextutorial9.html
http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Commands

There are a lot of things you can do without using LaTeX, such as exponents and subscripts. To get the extended menu (appears above the text input window), click Go Advanced below the text input window. Use the X2 button to write exponents and use the X2 to write subscripts.

Last edited by a moderator:
Thanks Mentallic! :)

No problem!

## 1. What does the limit of a cosine function represent?

The limit of a cosine function represents the value that the function approaches as the independent variable (usually denoted as x) gets closer and closer to a specific value, typically denoted as a. It is a fundamental concept in calculus and is used to analyze the behavior of a function at a particular point.

## 2. How is the limit of a cosine function calculated?

The limit of a cosine function can be calculated using the limit definition, which involves plugging in the specific value of a into the function and evaluating the resulting expression. Alternatively, it can also be calculated using algebraic manipulations or by applying known limit rules and properties.

## 3. Is the limit of a cosine function always defined?

No, the limit of a cosine function is not always defined. It depends on the behavior of the function at the specific value of a. If the function approaches a single value from both sides of a, then the limit exists and is equal to that value. However, if the function approaches different values from the left and right sides of a, then the limit does not exist.

## 4. How does the limit of a cosine function relate to continuity?

The limit of a cosine function is closely related to continuity. A function is continuous at a specific value of x if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, continuity and the existence of a limit are necessary conditions for each other.

## 5. Can the limit of a cosine function be used to find the derivative?

Yes, the limit of a cosine function can be used to find the derivative. In fact, the derivative of a cosine function is equal to the negative of the sine function, and this can be derived using the limit definition of a derivative. Therefore, understanding the limit of a cosine function is crucial in the study of calculus and derivatives.

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