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  • If  are the three roots of the equation <img alt="a x^3+b x^2+c x+

If \alpha, \beta, \gamma  are the three roots of the equation a x^3+b x^2+c x+d=0, a \neq 0 \: \; and\: \lim _{x \rightarrow p} f(x)=A, \lim _{x \rightarrow p} g(x)=B, \lim _{x \rightarrow p} h(x)=C, find the limit
\lim _{x \rightarrow p}\left(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right) \lim _{x \rightarrow p}\left[\frac{\alpha}{f(x)}+\frac{\beta}{g(x)}+\frac{\gamma}{h(x)}\right].

Option: 1

\left(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right)\left(\frac{\alpha}{A}+\frac{\beta}{B}+\frac{\gamma}{C}\right)


Option: 2

\left(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right)(\alpha A+\beta B+\gamma C)


Option: 3

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


Option: 4

\alpha \beta \gamma(A B+B C+C A)


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(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right) \lim _{x \rightarrow p}\left[\frac{\alpha}{f(x)}+\frac{\beta}{g(x)}+\frac{\gamma}{h(x)}\right]
=\lim _{x \rightarrow p}\left(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right)\left[\lim _{x \rightarrow p}\left\{\frac{\alpha}{f(x)}\right\}+\lim _{x \rightarrow p}\left\{\frac{\beta}{g(x)}\right\}+\lim _{x \rightarrow p}\left\{\frac{\gamma}{h(x)}\right\}\right]
=\left(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right)\left[\frac{\alpha}{\lim _{x \rightarrow p} f(x)}+\frac{\beta}{\lim _{x \rightarrow p} g(x)}+\frac{\gamma}{\lim _{x \rightarrow p} h(x)}\right]
=\left(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right)\left(\frac{\alpha}{A}+\frac{\beta}{B}+\frac{\gamma}{C}\right)



 

Posted by

seema garhwal

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