Exercise · ZW-MULTIVARIABLE-LIMIT-009 · ★★★ Advanced
For \(f(x,y)=\dfrac{xy^2}{x^2+y^4}\), the limit at \((0,0)\):
Choose one option.
Show answer
- D. Does not exist: it is \(0\) along lines and \(\dfrac12\) along \(x=y^2\)
Explanation
Along \(y=mx\) it tends to \(0\), but along the parabola \(x=y^2\) it equals \(\dfrac{y^4}{2y^4}=\dfrac12\). Testing only lines is misleading.
Review the concept: Limit of a function of several variables
The limit of a function \(f\) at a point \(\mathbf a\) is the number \(L\) that the values \(f(\mathbf x)\) approach as \(\mathbf x\) gets close to \(\mathbf a\) along any path.
\[\lim_{\mathbf x\to\mathbf a} f(\mathbf x)=L\]
More practice problems: limit of a function of several variables
- ZW-MULTIVARIABLE-LIMIT-001 Compute lim _(x,y)→(1,2)(x^2y+3).
- ZW-MULTIVARIABLE-LIMIT-002 Compute lim _(x,y)→(2,-1)(x+y)/(xy).
- ZW-MULTIVARIABLE-LIMIT-003 For f(x,y)=(xy)/(x^2+y^2), what value does f take along the line y=x (with x≠0)?
- ZW-MULTIVARIABLE-LIMIT-004 Compute lim _(x,y)→(0,0)(x^2y)/(x^2+y^2).
- ZW-MULTIVARIABLE-LIMIT-005 Compute lim _(x,y)→(0,0)(sin (x^2+y^2))/(x^2+y^2).
- ZW-MULTIVARIABLE-LIMIT-006 What is lim _(x,y)→(0,0)(x^3+y^3)/(x^2+y^2)?
- ZW-MULTIVARIABLE-LIMIT-007 If f has the same limit along two different paths to mathbf a, then:
- ZW-MULTIVARIABLE-LIMIT-008 What happens with lim _(x,y)→(0,0)(x^2-y^2)/(x^2+y^2)?