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Limits
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a limit is the value that a function or sequence "approaches" as the input approaches some value.
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• Fill in the blank….
You can always make a ___________ turn on red unless a sign is posted prohibiting it.
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Give me liberty or ___________ me death!
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Don’t cry over spilled ____________ .
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Limit – The most logical y-value for a given x-value. Example: look at the table of values to help us determine lim f(x) as x approaches 2.
f(x) = x f(x)
0 3
1 3
2 Undefined
3 3
4 3
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Let the graph below represent the function f(x)
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Two important ideas about limits:• The limit of a function at a point is the logical y-value at that point.
• We do not care what the value of a function actually is at the point where we’re looking for the limit.
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When g(x) = x + 1 and g(x) =
1lim ( )xg x
Find the
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Find
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Try on your own by sketching a graph
Find1) 2)
3)
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Three types of discontinuity
Removable discontinuity – When the graph is continuous except for a hole. Jump discontinuity – When the left and right side limits are different. Infinite discontinuity – Created by vertical asymptotes.
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Limits and Infinity
Cheat code: Divide each term by x of the highest degree in the function.
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Horizontal Asymptotes; Let f(x) = • The graph f(x) has, at most, one horizontal asymptote. – If the degree of the numerator (p(x)) is less than the degree of the denominator (q(x)), then the line y = 0 (the x-axis) is a horizontal asymptote.
– If the degree of p(x) is equal to the degree of q(x), then the line y = a/b, where a is the leading coefficient of p(x) an b is the leading coefficient of q(x).
– If the degree of p(x) > degree of q(x), then there are no horizontal asymptotes.
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Limits Practice with solutions
• http://archives.math.utk.edu/visual.calculus/1/limits.15/
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