Three limits: power, notable limit, hierarchy

Text

Evaluate the following limits. $\text{a)}\ \lim_{x\to1^+}x^{x}\qquad \text{b)}\ \lim_{x\to1^-}\frac{\sin(x-1)}{x-1}\qquad \text{c)}\ \lim_{x\to+\infty}\frac{3\ln x-x^2+2}{2x^2-3x^3+\cos x}$

Solution

a) The function $x^{x}=e^{x\ln x}$ is continuous for $x>0$; hence $\lim_{x\to1^+}x^{x}=1^{1}=1.$

b) Set $t=x-1\to0^-$: $\dfrac{\sin t}{t}\to 1$ (notable limit). So the limit is $1.$

c) Compare degrees: in the numerator the dominant term is $-x^2$ (degree $2$), in the denominator $-3x^3$ (degree $3$). Since the denominator grows faster, $\frac{-x^2}{-3x^3}=\frac{1}{3x}\to 0.$

$\boxed{a=1,\ b=1,\ c=0}$