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If \alpha ,\beta ,\gamma are the three roots of the equation ax^{3}+bx^{2}+cx+d= 0,a\neq 0 and \lim_{x\rightarrow p}f\left ( x \right )= A,\lim_{x\rightarrow p}g\left ( x \right )= B,\lim_{x\rightarrow p}h\left ( x \right )= C, find the limit  \lim_{x\rightarrow p}\left [ \alpha ^{2} f\left ( x \right )\left ( \beta +\gamma \right )+\beta ^{2}g\left ( x \right )\left ( \gamma +\alpha \right )+\gamma ^{2}h\left ( x \right )\left ( \alpha +\beta \right )\right ]. 

Option: 1

A\alpha ^{2}\left ( \beta +\gamma \right )+B\beta ^{2}\left ( \gamma +\alpha \right )+C\gamma ^{2}\left ( \alpha +\beta \right )


Option: 2

\left ( \alpha +\beta +\gamma \right )\left ( \alpha A^{2}+\beta B^{2}+\gamma C^{2} \right )


Option: 3

\left ( \alpha \beta \gamma \right )\left ( A^{3}+B^{3}+C^{3} \right )


Option: 4

\alpha \beta \gamma \left ( AB+BC+CA \right )


Answers (1)

best_answer

\alpha ,\beta ,\gamma are the provided three roots of the following equation.

ax^{3}+bx^{2}+cx+d= 0,a\neq 0

The given limits are

\lim_{x\rightarrow p}f\left ( x \right )= A \quad\cdots \left ( i \right )
\lim_{x\rightarrow p}g\left ( x \right )= B \quad\cdots \left ( ii \right )
\lim_{x\rightarrow p}h\left ( x \right )= C \quad\cdots \left ( iii \right )

Note the following algebraic rules for limits.
 

  • The “Sum law for limits” states that \lim_{x\rightarrow a}f\left ( x \right )+\lim_{x\rightarrow a}g\left ( x \right )= \lim_{x\rightarrow a}\left [ f\left ( x \right )+g\left ( x \right )\right ].
  • The “Constant multiple law for limits” indicates that \lim_{x\rightarrow a}cf\left ( x \right )= c\cdot \lim_{x\rightarrow a}f\left ( x \right )


So, applying the above laws for limits and using equations (i), (ii) and (iii), evaluate the following limit.

\lim _{x \rightarrow p}\left[\alpha^2 f(x)(\beta+\gamma)+\beta^2 g(x)(\gamma+\alpha)+\gamma^2 h(x)(\alpha+\beta)\right]
=\left[\alpha^2 \times \lim _{x \rightarrow p} f(x) \times(\beta+\gamma)+\beta^2 \times \lim _{x \rightarrow p} g(x) \times(\gamma+\alpha)+\gamma^2 \times \lim _{x \rightarrow p} h(x) \times(\alpha+\beta)\right]
=\alpha^2 A(\beta+\gamma)+\beta^2 B(\gamma+\alpha)+\gamma^2 C(\alpha+\beta)
=A \alpha^2(\beta+\gamma)+B \beta^2(\gamma+\alpha)+C \gamma^2(\alpha+\beta)

Posted by

sudhir.kumar

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