Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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Multiple Choice

Evaluate the following limit, if it exists. If the limit does not exist, select 'DNE'.
$lim_{(x, y) \to (0, 0)} \frac{x y}{x^{2} + y^{2}}$

DNE

0

\frac{1}{2}

1

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Verified step by step guidance

Step 1: Understand the problem. The goal is to evaluate the limit $ \lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2} $. This involves determining whether the value of the function approaches a single finite value, diverges, or depends on the path taken as $ (x, y) \to (0, 0) $.

Step 2: Analyze the function $ \frac{xy}{x^2 + y^2} $. Notice that the numerator $ xy $ involves the product of $ x $ and $ y $, while the denominator $ x^2 + y^2 $ is the sum of squares, which is always positive except at $ (0, 0) $.

Step 3: Test the limit along different paths. For example, substitute $ y = mx $ (a straight line path through the origin) into the function. This gives $ \frac{xy}{x^2 + y^2} = \frac{x(mx)}{x^2 + (mx)^2} = \frac{m x^2}{x^2 + m^2 x^2} = \frac{m}{1 + m^2} $. The result depends on $ m $, indicating the limit may depend on the path.

Step 4: Test another path, such as $ y = 0 $. Substituting $ y = 0 $ into the function gives $ \frac{xy}{x^2 + y^2} = \frac{x(0)}{x^2 + 0^2} = 0 $. This result is different from the previous path, further suggesting the limit depends on the path.

Step 5: Conclude that the limit does not exist (DNE). Since the value of $ \frac{xy}{x^2 + y^2} $ depends on the path taken as $ (x, y) \to (0, 0) $, the limit does not approach a single finite value. Therefore, the correct answer is DNE.