Practice section 1 of 8

Evaluate (a) $\lim_{x\to1}(x^2+4x-1)/(x+2)$, (b) $\lim_{x\to\infty}1/x$, and (c) $\lim_{x\to\infty}(3x-1)/(x+2)$.

Let $f:x\mapsto x^2$, $x\in\mathbb R$. Using the first principles of differentiation, show that $f^{\prime}(x)=2x$.

Differentiate $1/x^3-x^5+\sqrt{x^3}$ with respect to $x$.

Differentiate $1/(6x^2+4x+1)^2$ with respect to $x$.

Differentiate $(3x^2+1)(2x+7)^{1/2}$ with respect to $x$.

Differentiate $(2x^3+3)/(x^4+2)$ with respect to $x$.