Multivariable Calculus Equations

$\lim\limits_{x \to c} f(x) = L$

$\lim \limits_{x \to c} b = b$
${\mathop {\lim }\limits_{x \to c} x = c}$
${\mathop {\lim }\limits_{x \to c} kf(x) = k\cdot\mathop {\lim }\limits_{x \to c} f(x) = kL}$
${\mathop {\lim }\limits_{x \to c} [f(x) + g(x)] = \mathop {\lim }\limits_{x \to c} f(x) + \mathop {\lim }\limits_{x \to c} g(x) = L + M}$
${\mathop {\lim }\limits_{x \to c} [f(x) - g(x)] = \mathop {\lim }\limits_{x \to c} f(x) - \mathop {\lim }\limits_{x \to c} g(x) = L - M}$
${\mathop {\lim }\limits_{x \to c} [f(x)g(x)] = \mathop {\lim }\limits_{x \to c} f(x)\mathop {\lim }\limits_{x \to c} g(x) = LM}$
$\mathop {\lim }\limits_{x \to c} \frac{{f(x)}}{{g(x)}} = \frac{{\mathop {\lim }\limits_{x \to c} f(x)}}{{\mathop {\lim }\limits_{x \to c} g(x)}} = \frac{L}{M}$ provided $M \ne 0$

$\mathop {\lim }\limits_{x \to c} f{(x)^n} = {\left( {\mathop {\lim }\limits_{x \to c} f(x)} \right)^n} = {L^n} \hfill$
$g(x) \leqslant f(x) \leqslant h(x) \hfill$
$\mathop {\lim }\limits_{x \to c} g(x) = \mathop {\lim }\limits_{x \to c} h(x) = L \hfill$
$\mathop {\lim }\limits_{x \to c} f(x) = L \hfill$

$\mathop {\lim }\limits_{n \to \infty } ({a_n} + {b_n}) = \mathop {\lim }\limits_{n \to \infty } {a_n} + \mathop {\lim }\limits_{n \to \infty } {b_n} \hfill$
$\mathop {\lim }\limits_{n \to \infty } ({a_n} - {b_n}) = \mathop {\lim }\limits_{n \to \infty } {a_n} - \mathop {\lim }\limits_{n \to \infty } {b_n} \hfill$
$\mathop {\lim }\limits_{n \to \infty } c{a_n} = c\mathop {\lim }\limits_{n \to \infty } {a_n} \hfill$
$\mathop {\lim }\limits_{n \to \infty } c = c \hfill$
$\mathop {\lim }\limits_{n \to \infty } ({a_n}{b_n}) = \mathop {\lim }\limits_{n \to \infty } {a_n}\cdot\mathop {\lim }\limits_{n \to \infty } {b_n} \hfill$
$\mathop {\lim }\limits_{x \to \infty } \frac{{{a_n}}}{{{b_n}}} = \frac{{\mathop {\lim }\limits_{n \to \infty } {a_n}}}{{\mathop {\lim }\limits_{n \to \infty } {b_n}}} \hfill$ if $\mathop {\lim }\limits_{n \to \infty } {b_n} \ne 0$
$\mathop {\lim }\limits_{n \to \infty } a_n^p = {[\mathop {\lim }\limits_{n \to \infty } {a_n}]^p} \hfill$ if $p > 0$ and ${a_n}>0$

If $\mathop {\lim }\limits_{n \to \infty } \left| {{a_n}} \right| = 0 \hfill$, then $\mathop {\lim }\limits_{n \to \infty } {a_n} = 0$

If $\mathop {\lim }\limits_{n \to \infty } {a_n} = L$ and the function $f$ is continuous at $L$, then $\mathop {\lim }\limits_{n \to \infty } f({a_n}) = f(L)$

$\mathop {\lim }\limits_{n \to \infty } {r^n} = \left\{ {\begin{array}{*{20}{l}} 0&{ - 1 < r < 1} \\ 1&{r = 1} \end{array}} \right.$

The geometric series $\sum _{n=1}^{\infty } ar^{n-1}=a+ar+ar^2+\cdots$ is convergent if $\left| r \right| < 1$ and its sum is $\sum _{n=1}^{\infty } ar^{n-1}=\frac{a}{1-r}$

If $\left| r \right| \leqslant \ 1$, the geometric series is divergent.

If the series $\sum _{n=1}^{\infty }{a_n}$ is convergent, then $\mathop {\lim }\limits_{n \to \infty } {a_n} = 0$.

If $\mathop {\lim }\limits_{n \to \infty } {a_n}$ does not exist or if $\mathop {\lim }\limits_{n \to \infty } {a_n} \ne 0$, then the series $\sum _{n=1}^{\infty } {a_n}$ is divergent.

If $\sum{a_n}$ and $\sum {{b_n}}$ are convergent series, then so are the series $\sum {c{a_n}}$ (where $c$ is a constant), $\sum {({a_n} + {b_n})}$, and $\sum {({a_n} - {b_n})}$, and

$\sum \limits_{n=1}^{\infty} ca_n=c\sum\limits_{n=1}^{\infty } a_n$
$\sum _{n=1}^{\infty } (a_n+b_n)=\sum _{n=1}^{\infty } a_n+\sum _{n=1}^{\infty } b_n$
$\sum _{n=1}^{\infty } (a_n-b_n)=\sum _{n=1}^{\infty } a_n-\sum _{n=1}^{\infty } b_n$