Exercise · ZW-MULTIVARIABLE-LIMIT-007 · ★★☆ Intermediate
If \(f\) has the same limit along two different paths to \(\mathbf a\), then:
Choose one option.
Show answer
- D. One cannot conclude that the limit exists
Explanation
Agreeing along two paths is not enough: a third path may give another value. Only differing paths allow concluding that the limit does not exist.
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-008 What happens with lim _(x,y)→(0,0)(x^2-y^2)/(x^2+y^2)?
- ZW-MULTIVARIABLE-LIMIT-009 For f(x,y)=(xy^2)/(x^2+y^4), the limit at (0,0):