Multivariable Limits

Apr 2017
165
6
New York
Has anybody idea on what techniques I can apply on these limits?

at c and d I did direct substitution since there is no zero at the denominator. Is that correct?

a) I converted polar form and denominator became cos^2Theta+Sin^2theta = 1 so that limit exists

how about b?
no techniques I know work and I couldn't prove it DNE either.
 

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romsek

Math Team
Sep 2015
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1,674
USA
a) $\dfrac{x^4-y^4}{x^2+y^2} = \dfrac{(x^2-y^2)(x^2+y^2)}{x^2+y^2}=x^2-y^2$

Now just plug the values in.

b) $\lim \limits_{x\to 1\\y\to -1}~~\dfrac{xy}{x+y}$

at the limit values the numerator is finite and the denominator is 0 which suggest the limit is infinity. But by changing the direction you approach the limit point you can make the result head to either plus or minus infinity and thus the limit doesn't exist.

c) $\lim \limits_{x\to 1 \\y \to -1}~~\dfrac{x^2 ye^y}{x^4+4y^2}$

just plug the values in.

d) Same here, just plug the limit values in. I don't see why this problem would be harder. The graph looks perfectly smooth.
 
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