Page 53 - Calculus Workbook For Dummies

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Chapter 3: A Graph Is Worth a Thousand Words: Limits and Continuity

Solutions for Limits and Continuity

a At which of the following x-values are all three requirements for the existence of a limit satis-
fied, and what is the limit at those x-values? x = –2, 0, 2, 4, 5, 6, 8, 10, and 11.
At 0, the limit is 2.
At 4, the limit is 5.
At 8, the limit is 3.
At 10, the limit is 5.
To make a long story short, a limit exists at a particular x-value of a curve when the curve is
heading toward an exact y-value and keeps heading toward that y-value as you continue to zoom
in on the curve at the x-value. The curve must head toward that y-value from the right and from
the left (unless the limit is one where x approaches infinity). I emphasize “heading toward”
because what happens precisely at the given x-value isn’t relevant to this limit inquiry. That’s
why there is a limit at a hole like the ones at x = 8 and 10.

b For the rest of the x-values, state which of the three limit requirements are not satisfied. If one
or both one-sided limits exist at any of these x-values, give the value of the one-sided limit.
At –2 and 5, all three conditions fail.
At 2, 6, and 11, only the third requirement is not satisfied.
At 2, the limit from the left equals 5 and the limit from the right equals 3.
At 6, the limit from the left is 2 and the limit from the right is 3.
^ h
^ h
Finally, lim f x = 3 and lim f x = 5.
x " 11 - x " 11 +
c At which of the x-values are all three requirements for continuity satisfied?
The function in Figure 3-1 is continuous at 0 and 4. The common-sense way of thinking about
continuity is that a curve is continuous wherever you can draw the curve without taking your
pen off the paper. It should be obvious that that’s true at 0 and 4, but not at any of the other
listed x-values.
d For the rest of the x-values, state which of the three continuity requirements are not satisfied.
All listed x-values other than 0 and 4 are points of discontinuity. A discontinuity is just a high-
falutin’ calculus way of saying a gap. If you’d have to take your pen off the paper at some point
when drawing a curve, then the curve has a discontinuity there.
At 5 and 11, all three conditions fail.
At –2, 2, and 6, continuity requirements 2 and 3 are not satisfied.
At 10, requirements 1 and 3 are not satisfied.
At 8, requirement 3 is not satisfied.

^
^
e lim f xh does not exist (DNE) because there’s a vertical asymptote at –7. Or, because f xh
x " - 7
approaches 3- from the left and from the right, you could say the limit equals 3- .
f lim f xh does not exist because the limit from the left does not equal the limit from the
^
x " 5
right. Or you could say that the limit DNE because there’s a jump discontinuity at x = 5.
^ h
g lim f x = 5 because, like the second example problem, the limit at a hole is the height of
x " 18
^
the hole. The fact that f 18h is undefined is irrelevant to this limit question.