Chapter 2 | Limits
183
Example 2.28 Determining Continuity at a Point, Condition 3 Using the definition, determine whether the function f ( x ) = ⎧ ⎩ ⎨ sin x
x if x ≠0 1 if x =0
is continuous at x =0.
Solution First, observe that
f (0) =1.
Next,
sin x
f ( x ) = lim x →0
lim x →0
x =1.
Last, compare f (0) and lim x →1
f ( x ). We see that
f (0) =1= lim x →0 f ( x ). Since all three of the conditions in the definition of continuity are satisfied, f ( x ) is continuous at x =0.
Using the definition, determine whether the function f ( x ) = ⎧ ⎩
⎨ 2 x +1 if x <1 2 if x =1 − x +4 if x >1
2.21
is continuous at x =1.
If the function is not continuous at 1, indicate the condition for continuity at a point that fails to hold.
By applying the definition of continuity and previously established theorems concerning the evaluation of limits, we can state the following theorem.
Theorem 2.8: Continuity of Polynomials and Rational Functions Polynomials and rational functions are continuous at every point in their domains.
Proof Previously, we showed that if p ( x ) and q ( x ) are polynomials, lim x → a
p ( x ) = p ( a ) for every polynomial p ( x ) and
p ( x ) q ( x ) = p ( a ) q ( a ) as long as q ( a ) ≠0. Therefore, polynomials and rational functions are continuous on their domains.
lim x → a
□ We now apply Continuity of Polynomials and Rational Functions to determine the points at which a given rational function is continuous. Example 2.29 Continuity of a Rational Function
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