Theorem1.3.1Basic Limit Properties

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Let (btext{,}) (ctext{,}) (L) and (K) be real numbers, let (n) be a positive integer, and let (f) and (g) be functions with the following limits: begin{align*}lim_{xto c}f(x)amp=Lamplim_{xto c} g(x)amp = Ktext{.}end{align*}

1.3 Limits Graphical and Analytical Connections Remote Checklist. Sided limit of a real valued function. Formal definition of infinite limits, examples such as x x x 1 x 0 lim, 1 0 lim →+ →. (e) (i) Continuity of a real valued function in terms of limits and ∈−δ definition of continuity and two sided limits. Graphical representation of continuity of a real valued function.

The following limits hold.

(limlimits_{xto c} b = b)

(limlimits_{xto c} x = c)

(limlimits_{xto c}(f(x)pm g(x)) = Lpm K)

(limlimits_{xto c}(bcdot f(x)) = bL)

(limlimits_{xto c} (f(x)cdot g(x)) = LK)

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(limlimits_{xto c} (f(x)/g(x)) = L/Ktext{,}) when (Kneq 0)

(limlimits_{xto c} f(x)^n = L^n)

(limlimits_{xto c} sqrt[n]{f(x)} = sqrt[n]{L})

(If (n) is even, (L) must be non-negative.)

If either of the following holds:

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1. (limlimits_{xto c}f(x)=Ltext{,}) (limlimits_{xto L}g(x)=Ktext{,}) and (g(L)=K)

2. (limlimits_{xto c}f(x)=Ltext{,}) (limlimits_{xto L}g(x)=Ktext{,}) and (f(x)neq L) for all (x) close to but not equal to (c)

1.3 Limits Graphical And Analytical Connectionsap Calculus Answers

then (limlimits_{xto c}g(f(x)) = Ktext{.})