How to Show a Limit with More Than One Variable Does Not Exist

TL;DR

The limit does not exist when x and y approach 0 in the given function.

Transcript

hi in this problem we have a limit we have x y approaching 0 0 and the function is x squared over x squared plus y squared let's go ahead and work through it solution in order for a limit to exist must be equal to the same value no matter which way we approach so i am thinking that this limit is not going to exist because we can approach from diffe... Read More

Key Insights

• ⛔ Limits in calculus determine the value a function approaches when one or more variables approach a specific value.
• 💼 In this case, the limit does not exist because approaching from different directions gives different results.
• ⛔ Approaching from the y-axis yields a limit of 0, while approaching from the x-axis gives a limit of 1.
• ⛔ The existence of a limit requires the function to approach the same value regardless of the approach direction.
• ❣️ The function x squared over x squared plus y squared represents a relationship between two variables, x and y.
• ❣️ By setting x or y equal to 0, we can analyze the limit from specific directions.
• ❣️ The graphical representation of the problem helps visualize the limit approaches from the x-axis and y-axis.

Questions & Answers

Q: What is the function being analyzed for its limit?

The function is x squared over x squared plus y squared.

Q: Why does the limit not exist in this problem?

The limit does not exist because approaching from different directions (x-axis and y-axis) gives different answers.

Q: What is the limit when approaching from the y-axis?

When approaching from the y-axis (with x=0), the limit is 0.

Q: What is the limit when approaching from the x-axis?

When approaching from the x-axis (with y=0), the limit is 1.

Summary & Key Takeaways

• The limit of x squared over x squared plus y squared does not exist because approaching from different directions results in different answers.

• When approaching from the y-axis (with x=0), the limit is 0.

• When approaching from the x-axis (with y=0), the limit is 1.