Multivariable Limit Evaluator
The Multivariable Limit Evaluator computes the limit of a multivariable function as the variables approach a specified point, displaying computational steps with MathJax-rendered notation.
The Multivariable Limit Evaluator
The Multivariable Limit Evaluator computes the limit of a function \( f(x,y) \) as \((x,y)\) approaches a specified point, testing multiple paths to ensure consistency. Input the function and limit point, and see the computational steps with MathJax-rendered notation. Results are copyable, with sharing and embedding options for calculus students.
Example 1: The Multivariable Limit Evaluator for a Continuous Function
Function: \( f(x,y) = x^2 + y^2 \).
Limit Point: \( (0,0) \).
Step 1: Test along \( y = 0 \).
\( f(x,0) = x^2 \), limit as \( x \to 0 \): \( 0 \).
Step 2: Test along \( x = 0 \).
\( f(0,y) = y^2 \), limit as \( y \to 0 \): \( 0 \).
Step 3: Test along \( y = x \).
\( f(x,x) = x^2 + x^2 = 2x^2 \), limit as \( x \to 0 \): \( 0 \).
Step 4: Conclusion.
The limit is \( 0 \), as all paths agree.
Example 2: The Multivariable Limit Evaluator for a Non-Existent Limit
Function: \( f(x,y) = \frac{xy}{x^2 + y^2} \).
Limit Point: \( (0,0) \).
Step 1: Test along \( y = 0 \).
\( f(x,0) = \frac{x \cdot 0}{x^2 + 0} = 0 \), limit: \( 0 \).
Step 2: Test along \( y = x \).
\( f(x,x) = \frac{x \cdot x}{x^2 + x^2} = \frac{x^2}{2x^2} = \frac{1}{2} \), limit: \( \frac{1}{2} \).
Step 3: Conclusion.
The limit does not exist, as different paths yield different values.
Example 3: The Multivariable Limit Evaluator for a Rational Function
Function: \( f(x,y) = \frac{x^2 + y^2}{x + y} \).
Limit Point: \( (0,0) \).
Step 1: Test along \( y = 0 \).
\( f(x,0) = \frac{x^2}{x} = x \), limit as \( x \to 0 \): \( 0 \).
Step 2: Test along \( x = 0 \).
\( f(0,y) = \frac{y^2}{y} = y \), limit as \( y \to 0 \): \( 0 \).
Step 3: Test along \( y = x \).
\( f(x,x) = \frac{x^2 + x^2}{x + x} = \frac{2x^2}{2x} = x \), limit: \( 0 \).
Step 4: Conclusion.
The limit is \( 0 \), as all paths agree.
Example 4: The Multivariable Limit Evaluator for a Trigonometric Function
Function: \( f(x,y) = \frac{\sin(x^2 + y^2)}{x^2 + y^2} \).
Limit Point: \( (0,0) \).
Step 1: Use polar coordinates \( x = r \cos \theta \), \( y = r \sin \theta \).
\( f(r \cos \theta, r \sin \theta) = \frac{\sin(r^2)}{r^2} \), limit as \( r \to 0 \): \( 1 \).
Step 2: Conclusion.
The limit is \( 1 \), independent of \( \theta \).
Example 5: The Multivariable Limit Evaluator for a Complex Function
Function: \( f(x,y) = \frac{x^2 y}{x^4 + y^2} \).
Limit Point: \( (0,0) \).
Step 1: Test along \( y = 0 \).
\( f(x,0) = \frac{x^2 \cdot 0}{x^4 + 0} = 0 \), limit: \( 0 \).
Step 2: Test along \( y = x^2 \).
\( f(x,x^2) = \frac{x^2 \cdot x^2}{x^4 + x^4} = \frac{x^4}{2x^4} = \frac{1}{2} \), limit: \( \frac{1}{2} \).
Step 3: Conclusion.
The limit does not exist, as different paths yield different values.