Evaluate the following limits or determine that they do not exist. lim_{(x,y,z)rightarrow(2,2,3)}frac{x^2z-3x^2-y^2z+3y^2}{xz-3x-yz+3y}

sjeikdom0 2020-12-05 Answered
Evaluate the following limits or determine that they do not exist.
lim(x,y,z)(2,2,3)x2z3x2y2z+3y2xz3xyz+3y
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Expert Answer

StrycharzT
Answered 2020-12-06 Author has 102 answers
We have to evaluate the limits:
lim(x,y,z)(2,2,3)x2z3x2y2z+3y2xz3xyz+3y
Putting x=2, y=2 and z=3 in the function, we get
lim(x,y,z)(2,2,3)x2z3x2y2z+3y2xz3xyz+3y=22×33×2222×3+3×222×33×22×3+3×2
=121212+12666+6
=00
So it is indeterminate form of 00
Solving the limit by factorizing the numerator and denominator,
lim(x,y,z)(2,2,3)x2z3x2y2z+3y2xz3xyz+3y=lim(x,y,z)(2,2,3)x2(z3)y2(z3)x(z3)y(z3)
=lim(x,y,z)(2,2,3)(z3)(x2y2)(z3)(xy)
=lim(x,y,z)(2,2,3)(xy)(x+y)(xy)
=lim(x,y,z)(2,2,3)(x+y)
=2+2
=4
Hence, value of limit is 4.
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Jeffrey Jordon
Answered 2022-04-01 Author has 2047 answers

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