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}

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}

Question
Limits and continuity
asked 2020-12-05
Evaluate the following limits or determine that they do not exist.
\(\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{x^2z-3x^2-y^2z+3y^2}{xz-3x-yz+3y}\)

Answers (1)

2020-12-06
We have to evaluate the limits:
\(\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{x^2z-3x^2-y^2z+3y^2}{xz-3x-yz+3y}\)
Putting x=2, y=2 and z=3 in the function, we get
\(\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{x^2z-3x^2-y^2z+3y^2}{xz-3x-yz+3y}=\frac{2^2\times3-3\times2^2-2^2\times3+3\times2^2}{2\times3-3\times2-2\times3+3\times2}\)
\(=\frac{12-12-12+12}{6-6-6+6}\)
\(=\frac{0}{0}\)
So it is indeterminate form of \(\frac{0}{0}\)
Solving the limit by factorizing the numerator and denominator,
\(\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{x^2z-3x^2-y^2z+3y^2}{xz-3x-yz+3y}=\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{x^2(z-3)-y^2(z-3)}{x(z-3)-y(z-3)}\)
\(=\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{(z-3)(x^2-y^2)}{(z-3)(x-y)}\)
\(=\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{(x-y)(x+y)}{(x-y)}\)
\(=\lim_{(x,y,z)\rightarrow(2,2,3)}(x+y)\)
\(=2+2\)
\(=4\)
Hence, value of limit is 4.
0

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