186
Chapter 2 | Limits
x +2 x +1 =−∞
x +2 x +1 =+∞.
lim x →−1 −
lim x →−1 +
discontinuity, we must determine the limit at −1. We see that
and
Therefore, the function has an infinite discontinuity at −1.
⎧ ⎩ ⎨ x 2 if x ≠1 3 if x =1
2.23
For f ( x ) =
, decide whether f is continuous at 1. If f is not continuous at 1, classify the
discontinuity as removable, jump, or infinite.
Continuity over an Interval Now that we have explored the concept of continuity at a point, we extend that idea to continuity over an interval . As we develop this idea for different types of intervals, it may be useful to keep in mind the intuitive idea that a function is continuous over an interval if we can use a pencil to trace the function between any two points in the interval without lifting the pencil from the paper. In preparation for defining continuity on an interval, we begin by looking at the definition of what it means for a function to be continuous from the right at a point and continuous from the left at a point.
Continuity from the Right and from the Left A function f ( x ) is said to be continuous from the right at a if lim x → a +
f ( x ) = f ( a ).
A function f ( x ) is said to be continuous from the left at a if lim x → a −
f ( x ) = f ( a ).
A function is continuous over an open interval if it is continuous at every point in the interval. A function f ( x ) is continuous over a closed interval of the form ⎡ ⎣ a , b ⎤ ⎦ if it is continuous at every point in ( a , b ) and is continuous from the right at a and is continuous from the left at b . Analogously, a function f ( x ) is continuous over an interval of the form ( a , b ⎤ ⎦ if it is continuous over ( a , b ) and is continuous from the left at b . Continuity over other types of intervals are defined in a similar fashion. Requiring that lim x → a + f ( x ) = f ( a ) and lim x → b − f ( x ) = f ( b ) ensures that we can trace the graph of the function from the point ⎛ ⎝ a , f ( a ) ⎞ ⎠ to the point ⎛ ⎝ b , f ( b ) ⎞ ⎠ without lifting the pencil. If, for example, lim x → a + f ( x ) ≠ f ( a ), we would need to lift our pencil to jump from f ( a ) to the graph of the rest of the function over ( a , b ⎤ ⎦ .
Example 2.33 Continuity on an Interval State the interval(s) over which the function f ( x ) = x −1 x 2 +2 x
is continuous.
Solution Since f ( x ) = x −1 x 2 +2 x
is a rational function, it is continuous at every point in its domain. The domain of f ( x ) is the set (−∞, −2) ∪ (−2, 0) ∪ (0, +∞). Thus, f ( x ) is continuous over each of the intervals
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online