Limit as (x,y) --> (0,0) for sin(x^2+y^2)/(x^2+y^2)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find` ` lim (x,y)--> (0,0) [sin(x^2+y^2)/(x^2+y^2)]

Let x^2 + y^2 = t

=> `lim_(t-> 0) sin t/t`

Substituting t = 0 gives 0/0 which is indeterminate. Use l'Hopital's rule and substitute sin t and t by their derivatives

=> `lim_(t->0) cos t/1`

substituting t = 0 gives 1.

The required limit is 1.

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