Hi, I needed to know to to solve the limit of this equation analytically
lim
x->2
(4-X^2)/(3-√(X^2+5))
$\displaystyle \lim_{x\to{2}}\frac{4-x^2}{3-\sqrt{x^2+5}}$
The trick here is to multiply this fraction by the conjugate of the denominator:
$\displaystyle \begin{aligned}\lim_{x\to{2}}\frac{4-x^2}{3-\sqrt{x^2+5}}&=\lim_{x\to{2}}\frac{4-x^2}{3-\sqrt{x^2+5}}\cdot{\color{red}\frac{3+\sqrt{x^2+5} }{3+\sqrt{x^2+5}}}\\ &=\lim_{x\to{2}}\frac{(4-x^2)(3+\sqrt{x^2+5})}{9-(x^2+5)}\\ &=\lim_{x\to{2}}\frac{(4-x^2)(3-\sqrt{x^2+5})}{4-x^2}\\ &=\lim_{x\to{2}}(3+\sqrt{x^2+5})=\dots\end{aligned }$
Does this make sense?
--Chris