Overview of Indeterminate Forms

At x = a, if f(x) takes any of the forms \[\frac{0}{0}\], \[\frac{∞}{∞}\], 0 × ∞, 0°, 1^∞. 0^∞ and ∞⁰, then f(x) is said to be indeterminate at x = a. 

Other indeterminate forms are first reduced to the forms \[\frac{0}{0}\], \[\frac{∞}{∞}\]​.

• \[\lim_{x\to a}f(x)=0\mathrm{~and}\lim_{x\to a}g(x)=0\mathrm{,or"}\]

• \[“\lim_{x\to a}f(x)=\infty\mathrm{~and}\lim_{x\to a}g(x)=\infty,\]

• both f(x) and g(x) are differentiable in a neighbourhood of x = a,

• g′(x) ≠ 0

then \[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)}\]

provided the limit on the right-hand side exists.