Variable Change in Limits: When is it Permissible and Accurate?

In summary, when calculating limits using a change of variable, it is important to ensure that the change is made correctly and that all variables are replaced accurately. This method is allowed as long as the limit exists and the change is made accurately. Considering the epsilon-delta definition of limit can help in understanding why this method works.
  • #1
LeifEricson
11
0
Hello,

I see that a common method to calculating limits is a change of the variable. For example, to calculate:

[tex]\lim_{x \to \infty} \sin x \cdot \sin \frac {1}{x}[/tex]

We say that
[tex] t=\frac{1}{x} [/tex]

and then:

[tex]\lim_{x \to \infty} \sin x \cdot \sin \frac {1}{x} = \lim_{t \to 0^+} \sin \frac{1}{t} \cdot \sin t[/tex]

My question is:
When can we do that? When is it allowed? What are the conditions that this will be accurate and true? What do we have make sure before we can use that?

This is something I've pondered about for very long and haven't found an answer. I would really appreciate an explanation.

Thanks!
 
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  • #2
You just have to make sure that you are making the change correctly! If x= 1/t, or more generally, x= f(t), you just have to be sure you replace every "x" with f(t). The one point on which you have to be careful is replacing "[itex]x\rightarrow a[/itex]" with "[itex]t\rightarrow b[/itex]" where [itex]\lim_{t\rightarrow b} f(t)= a[/itex]- which means, of course, that limit must exist.
 
  • #3
You may want to consider the epsilon-delta definition of limit in a few cases to convince yourself of why it works.
 

Related to Variable Change in Limits: When is it Permissible and Accurate?

1. What is variable change in limits?

Variable change in limits, also known as the substitution method, is a technique used to evaluate limits in calculus. It involves substituting a new variable for the original variable in a limit expression and then simplifying the expression to determine the limit.

2. When should variable change in limits be used?

Variable change in limits should be used when evaluating a limit expression that involves a variable that approaches a certain value, such as infinity. It is also useful when the original variable in the expression is complicated or leads to an indeterminate form.

3. How is variable change in limits performed?

To perform variable change in limits, you first need to choose a new variable to substitute for the original variable. This new variable should be a function of the original variable and should simplify the expression. Then, replace all instances of the original variable with the new variable and simplify the expression to determine the limit.

4. What are the benefits of using variable change in limits?

Using variable change in limits can simplify the evaluation of limits and make them easier to solve. It can also help to avoid indeterminate forms and make the limit expression more manageable, especially when dealing with complex functions.

5. Are there any limitations to using variable change in limits?

While variable change in limits can be a powerful tool, it may not always work for all limit expressions. In some cases, it may not be possible to find a suitable substitution that simplifies the expression or leads to a meaningful result. It is important to carefully consider the expression before attempting to use this method.

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