q-Limits

List of limits when the variable q tends towards 1.

$\lim\limits_{q \to 1^{-}} \dfrac{1-q^n}{1-q} = n \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^n-1}{q-1} = n$

Hyperspecies

Limits related to the n-shifted and m-spaced Series of 1

$\lim\limits_{q \to 1^{-}} \dfrac{q^n}{1-q^{m+1}} = -\dfrac{1}{2} - \dfrac{n-m}{m+1} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^n}{q^{m+1}-1} = \dfrac{1}{2} + \dfrac{n-m}{m+1}$

Limits related to the n-shifted and n-spaced Series of 1

$\lim\limits_{q \to 1^{-}} \dfrac{q^n}{1-q^{n+1}} = -\dfrac{1}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^n}{q^{n+1}-1} = \dfrac{1}{2}$

Limits related to the n-shifted and (n–1)-spaced Series of 1

$\lim\limits_{q \to 1^{-}} \dfrac{q^n}{1-q^n} = -\dfrac{1}{2} - \dfrac{1}{n} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^n}{q^n-1} = \dfrac{1}{2} + \dfrac{1}{n}$

Limits related to the n-shifted Series of 1

$\lim\limits_{q \to 1^{-}} \dfrac{q^n}{1-q} = -\dfrac{1}{2} - n \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^n}{q-1} = \dfrac{1}{2} + n$

Limits related to the (n–1)-spaced Series of 1

$\lim\limits_{q \to 1^{-}} \dfrac{1}{1-q^n} = \dfrac{1}{2} - \dfrac{1}{n} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{q^n-1} = -\dfrac{1}{2} + \dfrac{1}{n}$

Superspecies

Limits related to the shifted and/or spaced Series of 1

$\lim\limits_{q \to 1^{-}} \dfrac{q}{1-q} = -\dfrac{3}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{q-1} = \dfrac{3}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{1-q} = -\dfrac{5}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{q-1} = \dfrac{5}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{1-q} = -\dfrac{7}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{q-1} = \dfrac{7}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{1-q} = -\dfrac{9}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{q-1} = \dfrac{9}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{1-q} = -\dfrac{11}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{q-1} = \dfrac{11}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{1-q^2} = 0 \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{q^2-1} = 0$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{1-q^2} = -\dfrac{1}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{q^2-1} = \dfrac{1}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{1-q^2} = -1 \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{q^2-1} = 1$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{1-q^2} = -\dfrac{3}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{q^2-1} = \dfrac{3}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{1-q^2} = -2 \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{q^2-1} = 2$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{1-q^2} = -\dfrac{5}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{q^2-1} = \dfrac{5}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{1-q^3} = \dfrac{1}{6} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{q^3-1} = -\dfrac{1}{6}$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{1-q^3} = -\dfrac{1}{6} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{q^3-1} = \dfrac{1}{6}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{1-q^3} = -\dfrac{1}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{q^3-1} = \dfrac{1}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{1-q^3} = -\dfrac{5}{6} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{q^3-1} = \dfrac{5}{6}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{1-q^3} = -\dfrac{7}{6} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{q^3-1} = \dfrac{7}{6}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{1-q^3} = -\dfrac{3}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{q^3-1} = \dfrac{3}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{1-q^4} = \dfrac{1}{4} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{q^4-1} = -\dfrac{1}{4}$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{1-q^4} = 0 \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{q^4-1} = 0$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{1-q^4} = -\dfrac{1}{4} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{q^4-1} = \dfrac{1}{4}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{1-q^4} = -\dfrac{1}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{q^4-1} = \dfrac{1}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{1-q^4} = -\dfrac{3}{4} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{q^4-1} = \dfrac{3}{4}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{1-q^4} = -1 \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{q^4-1} = 1$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{1-q^5} = \dfrac{3}{10} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{q^5-1} = -\dfrac{3}{10}$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{1-q^5} = \dfrac{1}{10} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{q^5-1} = -\dfrac{1}{10}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{1-q^5} = -\dfrac{1}{10} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{q^5-1} = \dfrac{1}{10}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{1-q^5} = -\dfrac{3}{10} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{q^5-1} = \dfrac{3}{10}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{1-q^5} = -\dfrac{1}{2} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{q^5-1} = \dfrac{1}{2}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{1-q^5} = -\dfrac{7}{10} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{q^5-1} = \dfrac{7}{10}$

Limits related to the shifted and/or spaced Series of Natural Numbers

$\lim\limits_{q \to 1^{-}} \dfrac{q}{(1-q)^2} = \dfrac{5}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{(q-1)^2} = \dfrac{5}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{(1-q)^2} = \dfrac{23}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{(q-1)^2} = \dfrac{23}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{(1-q)^2} = \dfrac{53}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{(q-1)^2} = \dfrac{53}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{(1-q)^2} = \dfrac{95}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{(q-1)^2} = \dfrac{95}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{(1-q)^2} = \dfrac{149}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{(q-1)^2} = \dfrac{149}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{(1-q^2)^2} = \dfrac{1}{24} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{(q^2-1)^2} = \dfrac{1}{24}$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{(1-q^2)^2} = -\dfrac{1}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{(q^2-1)^2} = -\dfrac{1}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{(1-q^2)^2} = \dfrac{1}{24} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{(q^2-1)^2} = \dfrac{1}{24}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{(1-q^2)^2} = \dfrac{5}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{(q^2-1)^2} = \dfrac{5}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{(1-q^2)^2} = \dfrac{25}{24} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{(q^2-1)^2} = \dfrac{25}{24}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{(1-q^2)^2} = \dfrac{23}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{(q^2-1)^2} = \dfrac{23}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{(1-q^3)^2} = \dfrac{5}{36} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{(q^3-1)^2} = \dfrac{5}{36}$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{(1-q^3)^2} = -\dfrac{1}{36} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{(q^3-1)^2} = -\dfrac{1}{36}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{(1-q^3)^2} = -\dfrac{1}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{(q^3-1)^2} = -\dfrac{1}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{(1-q^3)^2} = -\dfrac{1}{36} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{(q^3-1)^2} = -\dfrac{1}{36}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{(1-q^3)^2} = \dfrac{5}{36} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{(q^3-1)^2} = \dfrac{5}{36}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{(1-q^3)} = \dfrac{5}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{(q^3-1)^2} = \dfrac{5}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{(1-q^4)^2} = \dfrac{19}{96} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{(q^4-1)^2} = \dfrac{19}{96}$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{(1-q^4)^2} = \dfrac{1}{24} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{(q^4-1)^2} = \dfrac{1}{24}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{(1-q^4)^2} = -\dfrac{5}{96} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{(q^4-1)^2} = -\dfrac{5}{96}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{(1-q^4)^2} = -\dfrac{1}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{(q^4-1)^2} = -\dfrac{1}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{(1-q^4)^2} = -\dfrac{5}{96} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{(q^4-1)^2} = -\dfrac{5}{96}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{(1-q^4)^2} = \dfrac{1}{24} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{(q^4-1)^2} = \dfrac{1}{24}$

$\lim\limits_{q \to 1^{-}} \dfrac{1}{(1-q^5)^2} = \dfrac{71}{300} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{1}{(q^5-1)^2} = \dfrac{71}{300}$

$\lim\limits_{q \to 1^{-}} \dfrac{q}{(1-q^5)^2} = \dfrac{29}{300} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q}{(q^5-1)^2} = \dfrac{29}{300}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^2}{(1-q^5)^2} = -\dfrac{1}{300} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^2}{(q^5-1)^2} = -\dfrac{1}{300}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^3}{(1-q^5)^2} = -\dfrac{19}{300} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^3}{(q^5-1)^2} = -\dfrac{19}{300}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^4}{(1-q^5)^2} = -\dfrac{1}{12} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^4}{(q^5-1)^2} = -\dfrac{1}{12}$

$\lim\limits_{q \to 1^{-}} \dfrac{q^5}{(1-q^5)^2} = -\dfrac{19}{300} \quad \text{and} \quad \lim\limits_{q \to 1^{+}} \dfrac{q^5}{(q^5-1)^2} = -\dfrac{19}{300}$