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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 \beta \gamma \right )\lim_{x\rightarrow p}\left [ \alpha f\left ( x \right ) \times \beta g\left ( x \right )-\gamma h\left ( x \right )\right ].

Option: 1

\left ( \alpha \beta \gamma \right )\left ( \alpha \beta AB-\gamma C \right )


Option: 2

\left ( \alpha \beta \gamma \right )\left ( \alpha \beta AB\gamma C \right )


Option: 3

\left ( \alpha \beta \gamma \right )\left ( \alpha \beta +AB-\gamma C \right )


Option: 4

\left ( \alpha \beta \gamma \right )\left ( \alpha \beta +AB+\gamma C \right )


Answers (1)

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

Posted by

Ramraj Saini

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