Property 9
\(\samefrac00\text{ Uncertainty}\)
\(\lim\limits_{x\to a}\samefrac{P(x)}{Q(x)}=\samefrac00\)
\(P(x)\) and \(Q(x)\) are factorized and the factors that make the equation zero get cancelled.
\(\lim\limits_{x\to a}\samefrac{\cancel{(x-a)}\cdot P_1(x)}{\cancel{(x-a)}\cdot Q_1(x)}=\lim\limits_{x\to a}\samefrac{P_1(x)}{Q_1(x)}\)
1.
\(\lim\limits_{x\to1}\left(\samefrac{x^2-1}{x-1}\right)=?\)
2.
\(\lim\limits_{x\to2}\left(\samefrac{x^2-3x+2}{x-2}\right)=?\)
3.
\(\lim\limits_{x\to-1}\left(\samefrac{x^2+3x+2}{x^2-1}\right)=?\)
4.
\(\lim\limits_{x\to1}\left(\samefrac{x^3-1}{x^2-1}\right)=?\)
5.
\(\lim\limits_{x\to2^+}\left(\samefrac{x-2}{|x^2-4|}\right)=?\)
6.
\(k\in\mathbb R\)
\(\lim\limits_{x\to1}\left(\samefrac{x^2+ax-4}{x^2-1}\right)=k\)
\(\Rightarrow a=?\)
7.
\(k\in\mathbb R\)
\(\lim\limits_{x\to2}\left(\samefrac{x^2+ax+6}{x^2-3x+2}\right)=k\)
\(\Rightarrow a=?\)
Property 10
\(\samefrac00\text{ Uncertainty}\)
In radical expressions if there is 0/0 uncertainty both parts of the fraction is to be multiplied by the conjugate.
Example
\(\lim\limits_{x\to3}\left(\samefrac{x-3}{\sqrt x-\sqrt3}\right)=?\)
Answer
\(\lim\limits_{x\to3}\left(\samefrac{x-3}{\sqrt x-\sqrt3}\right)=\lim\limits_{x\to3}\samefrac{(x-3)(\sqrt x+\sqrt3)}{(\sqrt x-\sqrt3)(\sqrt x+\sqrt3)}\)
\(=\lim\limits_{x\to3}\samefrac{\cancel{(x-3)}(\sqrt x+\sqrt3)}{\cancel{(x-3)}}\)
\(=\lim\limits_{x\to3}(\sqrt x+\sqrt3)\)
\(=2\sqrt3\)
1.
\(\lim\limits_{x\to4}\left(\samefrac{x-4}{\sqrt x-2}\right)=?\)
2.
\(\lim\limits_{x\to2}\left(\samefrac{x^2-4}{\sqrt{x+7}-3}\right)=?\)
3.
\(\lim\limits_{x\to3}\left(\samefrac{x-3}{\sqrt{x+1}-2}\right)=?\)
4.
\(\lim\limits_{x\to1}\left(\samefrac{\sqrt{x+3}-2}{\sqrt x-1}\right)=?\)