Instructions

For the following questions answer them individually

Question 41

Question 42

Question 43

Question 44

Question 45

Question 46

Question 47

Question 48

# Let $$\alpha(a)$$ and $$\beta(a)$$ be the roots of the equation $$\left(\sqrt[3]{1 + a} - 1\right)x^2 + \left(\sqrt{1 + a} - 1\right)x + \left(\sqrt[6]{1 + a} - 1\right) = 0$$ where a > -1. Then $$\lim_{a \rightarrow 0^+}\alpha(a)$$ and $$\lim_{a \rightarrow 0^+}\beta(a)$$ are

Instructions

Let $$f(x) = (1 - x)^2 \sin^2 x + x^2$$ for all $$x \in IR$$ and let $$g(x) = \int_{1}^{x}\left(\frac{2(t - 1)}{t + 1} - \ln t\right)f(t) dt$$ for all $$x \in (1, \infty)$$.

Question 49

Question 50

OR