Answer to Question #177370 in Calculus for Moel Tariburu

Question

Evaluate the derivative lim┬(x→1)⁡[1/(1-x) ∫_(x^2)^x▒(t^2-1)dt]

Answer 1

$lim_{x\rightarrow1}[\dfrac{1}{1-x}\int_{-x^2}^{x^2}(t^2-1)dt]$

$=lim_{x\rightarrow1}[\dfrac{1}{1-x}(\dfrac{t^3}{3}-t)|_{-x^2}^{x^2}$

$=lim_{x\rightarrow1}[\dfrac{1}{1-x}(\dfrac{x^6}{3}-x^2+\dfrac{x^6}{3}-x^2)$

$=lim_{x\rightarrow1}[\dfrac{1}{1-x}2x^2(\dfrac{x^3}{3}-1)$

$=\dfrac{2(1)^2(\dfrac{1}{3}-1)}{1-1}=\infty$

Hence The given limit does not exist.