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

About russjohnson09

I am a student at GVSU.
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