"Does lim(x,y)→(3,−1)(x−3)(y+1)(x−3)2+(y+1)2 exist?" was answered by our community

Carol Gates

Answered question

2021-10-25

Does

lim_{(x, y) \to (3, - 1)} \frac{(x - 3) (y + 1)}{{(x - 3)}^{2} + {(y + 1)}^{2}}

exist?

jlo2niT

Skilled2021-10-26Added 96 answers

Given that

lim_{(x, y) \to (3, - 1)} \frac{(x - 3) (y + 1)}{{(x - 3)}^{2} + {(y + 1)}^{2}}

The value of the limit

lim_{(x, y) \to (3, - 1)} \frac{(x - 3) (y + 1)}{{(x - 3)}^{2} + {(y + 1)}^{2}}

do not exist as the denominator becomes 0.
Evaluate

lim_{(x, y) \to (3, - 1)} \frac{(x - 3) (y + 1)}{{(x - 3)}^{2} + {(y + 1)}^{2}}

as shown below

lim_{(x, y) \to (3, - 1)} \frac{(x - 3) (y + 1)}{{(x - 3)}^{2} + {(y + 1)}^{2}} = \frac{(0 - 3) (0 + 1)}{{(0 - 3)}^{2} + {(0 + 1)}^{2}}

= \frac{(- 3) (1)}{9 + 1}

= - \frac{3}{10}

Therefore, the limits when x=0 and y=0

lim_{(x, y) \to (3, - 1)} \frac{(x - 3) (y + 1)}{{(x - 3)}^{2} + {(y + 1)}^{2}} = - \frac{3}{10}

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