Indeterminate forms:

“∞ − ∞”  “∞ · 0”  “\frac{∞}{∞}”  “\frac{0}{0}”
“∞^{0}”  “0^{0}”  “1^{∞}”

L’Hospital’s Rule: Suppose f′ and g′ exist, g′(x) ≠ 0 near a except possibly at a where a ∈ R ∩ {+∞, −∞} and if either 1.) lim_{x→a}f(x) = lim_{x→a}g(x) = 0 or 2.) lim_{x→a}f(x) = ±∞ and lim_{x→a}g(x) = ±∞ and if lim_{x→a} \frac{f′(x)}{g′(x)} exists, then

lim_{x→a} \frac{f(x)}{g(x)} = lim_{x→a} \frac{f′(x)}{g′(x)}

Ex 1) lim_{x→+∞} \frac{−2x^{2}+3}{5x^{2}+4x}

Old method: lim_{x→+∞} \frac{−2x^{2}+3}{5x^{2}+4x} = lim_{x→+∞} \frac{x^{2}(−2+ \frac{3}{x^{2}} )}{x^{2}(5+ \frac{4}{x})} = New method: lim_{x→+∞} \frac{−2x^{2}+3}{5x^{2}+4x}

lim_{x→1} \frac{1−x+ln(x)}{x^{3}−3x+2}