Find the limit if it exists. If the limit does not exist, state the reason. ...

Question:

Find the limit if it exists. If the limit does not exist, state the reason.

{eq}\lim_{(x,y)\rightarrow (0,0)}\frac{x^2y}{x^4 + y^2} {/eq}

Double Limit of Function of two variables:

Let us consider any function {eq}z=f(x,y) {/eq} and let

1) Path 1:

{eq}L_1=\mathop {\lim }\limits_{x \to 0} \mathop {\lim }\limits_{y \to 0} f(x,y) {/eq}

2)Path 2:

{eq}L_2=\mathop {\lim }\limits_{y \to 0} \mathop {\lim }\limits_{x \to 0} f(x,y) {/eq}

3)Path 3: (Along {eq}{y = m{x^n}} {/eq})

{eq}{L_3} = \mathop {\lim }\limits_{x \to 0} \mathop {\lim }\limits_{y = m{x^n}} f(x,y) {/eq}

Then the double limit {eq}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( 0,0 \right)} f(x,y) {/eq} exist if {eq}L_1=L_2=L_3 {/eq}

Answer and Explanation:

Here the given function is:{eq}f(x,y) = \frac{x^2y}{x^4 + y^2} {/eq}

Path 1:

{eq}\lim_{(x,y) \rightarrow (0,0)} f(x,y) {/eq} along the path {eq}y = 0 {/eq} is given by:

{eq}{L_1} = \mathop {\lim }\limits_{x \to 0} \mathop {\lim }\limits_{y \to 0} \left( {\frac{{{x^2}y}}{{{x^4} + {y^2}}}} \right) = \mathop {\lim }\limits_{x \to 0} \frac{0}{{{x^4}}} = 0 {/eq}

Path 2:

{eq}\lim_{(x,y) \rightarrow (0,0)} f(x,y) {/eq} along the path {eq}x = 0 {/eq} is given by:

{eq}{L_2} = \mathop {\lim }\limits_{y \to 0} \mathop {\lim }\limits_{x \to 0} \left( {\frac{{{x^2}y}}{{{x^4} + {y^2}}}} \right) = \mathop {\lim }\limits_{y \to 0} \frac{0}{{{y^2}}} = 0 {/eq}

Path 3:

{eq}\lim_{(x,y) \rightarrow (0,0)} f(x,y) {/eq} along the path {eq}y = mx^2 {/eq} is given by:

{eq}{L_3} = \mathop {\lim }\limits_{x \to 0} \mathop {\lim }\limits_{y = m{x^2}} \left( {\frac{{{x^2}y}}{{{x^4} + {y^2}}}} \right) = \mathop {\lim }\limits_{x \to 0} \frac{{\left( {m{x^2}} \right)\left( {{x^2}} \right)}}{{{x^4} + {m^2}{x^4}}} = \mathop {\lim }\limits_{x \to 0} \frac{{\left( {m{x^4}} \right)}}{{{x^4}\left( {1 + {m^2}} \right)}} = \mathop {\lim }\limits_{x \to 0} \frac{m}{{(1 + {m^2})}} = \frac{m}{{(1 + {m^2})}} = [{\text{Depends on m}}] {/eq}

Hence the limit doesn't exist.


Learn more about this topic:

How to Determine the Limits of Functions

from Math 104: Calculus

Chapter 7 / Lesson 4
12K

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