Properties of the limits

Here a list of the properties of the limiting in differential Calculus.

Property of the constant function

$\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; B\; =\; b$

Approaches graphic

The graphics of the function F defined by $f\; (X)\; =\; b$ are a line of equation $y\; =\; b$, the limit of the function is the ordinate in the beginning.

Property of the function F defined by $f\; (X)\; x$

$\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; X\; =\; a$

Approaches graphic

The graph of this function is a line passing by the origin, of equation $y\; =\; x$. The limit when X approaches $a$, corresponds to the ordinate of the point of X-coordinate $a$ on the line, this limit is worth thus $a$.

Property of the multiplication by a constant

If $f$ admits in $a$ a finished limit and if D is a real constant then the function $d\; \backslash \; times\; f$ admits a limit in $a$ such as:

$\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; F\; (X)\; =\; D\; \backslash \; times\; \backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; F\; (X)$

the limit of a function multiplied by a constant is equal to the constant multiplied by the limit of the function.

Regulate sum

If the functions $f$ and $g$ admit each one a limit finished in $a$, then the function $f+g$ admits it also a limit in $a$ such as:

$\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; +\; G\; (X)\; =\; \backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; F\; (X)\; +\; \backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; G\; (X)$

the limit of a sum is equal to the sum of the limits.

Regulate difference

If the functions $f$ and $g$ admit each one a limit finished in $a$, then the function $f-g$ admits it also a limit in $a$ such as:

$\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; -\; G\; (X)\; =\; \backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; F\; (X)\; -\; \backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; G\; (X)$

the limit of a difference is equal to the difference in the limits.

Regulate product

If the functions $f$ and $g$ admit each one a limit finished in $a$, then the function $f\; \backslash \; times\; g$ admits it also a limit in $a$ such as:

$\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; G\; (X)\; =\; \backslash \; to\; has\}\; F\; (X)\; \backslash \; to\; has\}\; G\; (X)$

the limit of a product is equal to the product of the limits.

Regulate quotient

If the function $f$ admits a limit finished in $a$ and if the function $g$ admits a nonnull limit finished in $a$, then the function $\backslash \; frac\; \{F\}\; \{G\}$ admits it also a limit in $a$ such as:

$\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; \backslash \; frac\; \{F\; (X)\}\{G\; (X)\}\; =\; \backslash \; frac\; \{\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; F\; (X)\}\{\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; G\; (X)\}$

the limit of a quotient is equal to the quotient of the limits (if the denominator is not null).

Regulate powers

If $f$ admits in $a$ a limit finished then the function $X\; \backslash \; rightarrow\; ^n$ admits also a limit in $a$ such as: $\backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; ^n\; =\; \backslash \; to\; has\}\; F\; (X)\; ^n$

the limit of a function raised with the power $n\; (\backslash \; in\; \backslash \; NR)$ is equal to the limit of the function, high with the power $n$.

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