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  • If are the three roots of the equation <img alt="ax^{2}+bx^{2}+cx+d= 0,

If \alpha ,\beta ,\gamma are the three roots of the equation ax^{2}+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 +\beta +\gamma \right )\lim_{x\rightarrow p}\left [ \alpha f\left ( x \right )+\beta g\left ( x \right )+\gamma h\left ( x \right ) \right ].

Option: 1

\left ( \alpha +\beta +\gamma \right )\left ( \alpha A+\beta B+\gamma C \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 )
  • The “Product law for limits” states that \lim_{x\rightarrow a}f\left ( x \right )\times \lim_{x\rightarrow a}g\left ( x \right )=\lim_{x\rightarrow a} \left [ f\left ( x \right )\times g\left ( x \right )\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 +\beta +\gamma \right )\lim_{x\rightarrow p}\left [\alpha f\left ( x \right ) +\beta g\left ( x \right )+\gamma h\left ( x \right )\right ]
= \lim_{x\rightarrow p}\left ( \alpha +\beta +\gamma \right )\left [\alpha \lim_{x\rightarrow p} f\left ( x \right ) +\beta \lim_{x\rightarrow p} g\left ( x \right )+\gamma\lim_{x\rightarrow p} h\left ( x \right ) \right ]
=\left ( \alpha +\beta +\gamma \right )\left ( \alpha A+\beta B+\gamma C \right )

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