How Do You Solve Double Limits Involving Two Variables?

  • Thread starter utkarshakash
  • Start date
  • Tags
    Limits
In summary, a double limit is a mathematical concept that represents the behavior of a function as two variables approach a specific point simultaneously. To find a double limit, you need to first determine the limit of the function as each variable approaches the specified point separately and then combine these two limits. The difference between a single and a double limit is that a single limit involves one variable approaching a point while a double limit involves two variables approaching a point. The properties of double limits include the sum, product, quotient, and composition properties, which simplify the process of evaluation. L'Hopital's rule should be used when evaluating a double limit that results in an indeterminate form.
  • #1
utkarshakash
Gold Member
854
13

Homework Statement


[itex]\stackrel{lim}{y→0}\left( \stackrel{lim}{x→∞} \dfrac{\left( 1+\dfrac{ay}{x} \right)^x - \left( 1+\dfrac{by}{x} \right)^x}{y} \right)[/itex]

Homework Equations



The Attempt at a Solution


I need some hints. I really don't know how to solve these kinds of limits in which two variables are involved.
 
Physics news on Phys.org
  • #2
Just do the inner one first with y held constant, then do the outer one. If you know how to do one-dimensional limits then these are just as easy!
 
  • #3
You need to know that [itex]\lim_{x\to \infty}\left(1+ \frac{a}{x}\right)^x=e^a[/itex]
 

FAQ: How Do You Solve Double Limits Involving Two Variables?

What is a double limit?

A double limit is a mathematical concept that represents the behavior of a function as two variables approach a specific point simultaneously. It is the limit of a function in two directions.

How do I find a double limit?

To find a double limit, you need to first determine the limit of the function as each variable approaches the specified point separately. Then, you can combine these two limits to find the double limit.

What is the difference between a single and a double limit?

A single limit involves the behavior of a function as one variable approaches a specific point, while a double limit involves the behavior of a function as two variables approach a specific point simultaneously.

What are the properties of double limits?

The properties of double limits include the sum property, product property, quotient property, and composition property. These properties allow us to simplify the process of evaluating double limits.

When should I use L'Hopital's rule to find a double limit?

L'Hopital's rule should be used when evaluating a double limit that results in an indeterminate form, such as 0/0 or ∞/∞. This rule allows us to find the limit by taking the derivatives of the numerator and denominator and evaluating the resulting limit.

Similar threads

Solve the problem involving the given double integral
Replies
1
Views
928
Find the volume of the solid bound by the three coordinate planes
Replies
2
Views
852
Solve the problem involving space curve
Replies
2
Views
707
Solve the given parametric equation
Replies
3
Views
1K
Solve the given first order differential equation
Replies
8
Views
1K
Solve the attached problem that involves circle and tangent
Replies
2
Views
872
Solve this problem that involves implicit differentiation
Replies
15
Views
1K
Solve this problem that involves a probability density function
Replies
8
Views
1K
Find Limit: Homework Statement & Equations
Replies
12
Views
2K
Solve the given equation using the Lambert W function
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top