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  • If <img alt="\lim_{p\rightarrow a}\left ( x+y+z \right )= L,\lim_{p\rightarrow a}\left ( xy+yz+zx \right )=M," src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim_%7Bp%5Crightarrow%20a%7D%5

If \lim_{p\rightarrow a}\left ( x+y+z \right )= L,\lim_{p\rightarrow a}\left ( xy+yz+zx \right )=M,  find the value of \lim_{p\rightarrow a}\left ( x^{3}+y^{3}+z^{3}-3xyz \right )

Option: 1

L^{3}+3LM


Option: 2

L^{3}-3LM


Option: 3

M^{3}-3LM


Option: 4

3LM


Answers (1)

best_answer

The following are the noteworthy laws applicable in the algebraic calculations for limits.

  • The “Difference law for limits” indicates that \lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)= \lim _{x \rightarrow a} \left [ f\left ( x \right )-g\left ( x \right ) \right ].
  • The “Product law for limits” shows that \lim _{x \rightarrow a} f(x)\times\lim _{x \rightarrow a} g(x)= \lim _{x \rightarrow a} \left [ f\left ( x \right )\times g\left ( x \right ) \right ].
  • The “Power law for limits” describes that \lim _{x \rightarrow a}\left ( f(x) \right )^{n}= \left ( \lim _{x \rightarrow a}f\left ( x \right ) \right )^{n} when n= +ve integer.


The provide limits are
\lim _{p \rightarrow a}\left ( x+y+z \right )=L \quad \left ( i \right )
\lim _{p \rightarrow a}\left ( xy+yz+zx \right )=M \quad \left ( ii \right )

Algebraically, the expression x^{3}+y^{3}+z^{3}-3xyz can be written in the following way.
x^{3}+y^{3}+z^{3}-3xyz= \left ( x+y+z \right )^{3}-3\left ( x+y+z \right )\left ( xy+yz+zx \right )\quad \left ( iii \right )

Now, use the equations (i), (ii) and (ii) to derive the following.

\lim _{p \rightarrow a}\left ( x^{3}+y^{3}+z^{3}-3xyz \right )
= \lim _{p \rightarrow a}\left [ \left ( x+y+z \right )^{3}-3\left (x+y+z \right ) \left ( xy+yz+zx \right )\right ]
= \lim _{p \rightarrow a} \left ( x+y+z \right )^{3}-3\lim _{p \rightarrow a} \left (x+y+z \right ) \times\lim _{p \rightarrow a} \left ( xy+yz+zx \right )
=\left [ \lim _{p \rightarrow a}\left ( x+y+z \right ) \right ]^{3}-3 \lim _{p \rightarrow a}\left ( x+y+z\right )\times \lim _{p \rightarrow a} \left ( xy+yz+zx \right )
=L^{3}-3LM

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