Limits Question 6
Question 6 - 2024 (29 Jan Shift 2)
Let the slope of the line $45 x+5 y+3=0$ be $27 r_{1}+\frac{9 r_{2}}{2}$ for some $r_{1}, \quad r_{2} \in R$. Then
$\operatorname{Lim}{x \rightarrow 3}\left(\int{3}^{x} \frac{8 t^{2}}{\frac{3 r_{2} x}{2}-r_{2} x^{2}-r_{1} x^{3}-3 x} d t\right)$ is equal to
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Answer (12)
Solution
According to the question,
$27 r_{1}+\frac{9 r_{2}}{2}=-9$
$\lim {x \rightarrow 3} \frac{\int{3}^{x} 8 t^{2} d t}{\frac{3 r_{2} x}{2}-r_{2} x^{2}-r_{1} x^{3}-3 x}$
$=\lim {x \rightarrow 3} \frac{8 x^{2} \text { mathon }}{\frac{3 r{2}^{2}}{2}-2 r_{2} x-3 r_{1} x^{2}-3}$ (using LH’ Rule)
$=\frac{\text { mathon } 72}{\frac{3 r_{2}}{2}-6 r_{2}-27 r_{1}-3}$
$=\frac{\text { math } 72}{-\frac{9 r_{2}}{2}-27 r_{1}-3}$
$=\frac{72}{9-3}=12$
