Determine whether the limit converges or diverges. If it converges, what does it converge to?
Answer
converges to 0.
diverges.
converges to 1.
Find the value of each integral. Use a calculator if you want, and give each answer as a decimal.
(a)
(b)
(c)
Does converge or diverge? If it converges, what does it converge to?
Look at the sequence of integrals
as b gets larger:
These values are getting closer and closer to 1.
As b approaches ∞,
approaches 1. This means
Each integral in the above exercise is the area between the graph of and the x-axis on an interval with left endpoint 1.
As b approaches ∞, the area
approaches the total area between the graph of
and the x-axis on the interval [1,∞).
In symbols, the total area between and the x-axis on [1,∞) is
We abbreviate this limit by writing
a.
b.
c.
as the lower limit of integration b gets closer to 0:
These values are getting closer to 2.
As b approaches 0, approaches 2. So
converges to 2.
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converges to 0.
diverges.
converges to 1.
converge or diverge? If it converges, what does it converge to?
and the x-axis on an interval with left endpoint 1. 
and the x-axis on [1,∞) is
converge or diverge? If it converges, what does it converge to?
approaches 2. So