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Definition
| laws that help you solve limits algebraically rather than with a graph, calculator, or chart/table |
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| Limit Law: lim x>a [f(x) + g(x)] |
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Definition
| lim x>a f(x) + limx>a g(x) |
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| Limit Law: lim x>a [f(x) - g(x)] |
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Definition
| lim x>a f(x) - lim x>a g(x) |
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| Limit Law: lim x>a [cf(x)] |
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| Limit Law: lim x>a [f(x) * g(x)] |
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Definition
| lim x>a f(x) * lim x>a g(x) |
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| Limit Law: lim x>a [f(x)/g(x)] |
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Definition
| lim x>a f(x) / lim x>a g(x) |
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| Limit Law: lim x>a [f(x)]^n |
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| Limit Law: lim x>a (n)√(f(x)) |
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| What is the Direct Substitution Property (DSP)? |
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Definition
If "f" is a polynomial or rational function & "a" is in the domain of "f," then lim x>a f(x) = f(a).
You can use plug-in for polynomials & rational functions. |
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| What is the Definition of a Vertical Asymptote? |
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Definition
1. lim x>a f(x) = +/- ∞
2. lim x>a- f(x) = +/- ∞
3. lim x>a+ f(x) = +/- ∞
x=a is a vert. asymp. if x=a causes division by 0 after canceling common factors |
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| What is the Theorem of Limits if f(x) ≤ g(x)? |
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Definition
| If f(x) ≤ g(x) when x is near "a" (except possibly at "a") & the limits of both f(x) & g(x) exist as "x" approaches "a," then lim x>a f(x) ≤ lim x>a g(x). |
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| What is the Squeeze Theorem? |
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Definition
| If f(x) ≤ h(x) ≤ g(x) when "x" is near "a" (except possible at "a") & not only do the limits of the functions exist, & lim x>a f(x) = lim x>a h(x) = L, we know that lim x>a g(x) = L also. |
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| What is the Definition of Continuity? |
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Definition
A function "f" is continuous at a number if lim x>a f(x) = f(a). Basically- continuous if:
1. f(a) is defined
2. lim x>a f(x) exists
3. lim x>a f(x) = f(a) |
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| What the Definition of a Jump Discontinuity? |
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Definition
| When lim x>a- f(x) exists & lim x>a+ exists but they are not equal. |
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| What is the Definition of a Removable Discontinuity? |
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| When lim x>a f(x) ≠ f(a). |
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| Theorem: If "f" & "g" are continuous at "a," and "c" is a constant: |
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Definition
| All the limit laws can be applied, giving a continuous function. |
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| Theorem: regarding polynomials & continuity? |
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Definition
| A polynomial is continuous on the domain (-∞, +∞). |
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| Theorem: regarding rational functions and continuity? |
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Definition
| A rational function is continuous for its domain. |
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| Theorem: absolute value function & continuity? |
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Definition
| An absolute value function is continuous everywhere. |
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| Theorem: regarding nth root function [f(x) = (n)√(x)]? |
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Definition
- if "n" is odd, then f(x) is continuous over (-∞, ∞)
- if "n" is even, then f(x) is continuous at positive numbers |
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| Theorem: regarding trigonometric functions and continuity? |
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| Trigonometric functions are continuous on their domains (as long as you don't divide by zero). |
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| What is the domain of cosx? |
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Definition
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| What is the domain of sinx |
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| What is the domain of tanx? |
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Definition
| x ≠ π/2 + kπ, where "k" is an integer |
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| What is the domain of secant? |
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Definition
| x ≠ π/2 + kπ and "k" is an integer |
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| What is the domain of cotx? |
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| What is the domain of cscx? |
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