Theory and Examples

Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

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Recall the definition of an odd function: A function f(x) is odd if for all x in the domain of f, f(-x) = -f(x).

Consider the given limit: lim as x approaches 0 from the positive side (x → 0+) of f(x) is 3. This means as x gets closer to 0 from the right, f(x) approaches 3.

Use the property of odd functions: Since f is odd, f(-x) = -f(x). Therefore, if x approaches 0 from the positive side, -x approaches 0 from the negative side.

Apply the odd function property to the limit: lim as x approaches 0 from the negative side (x → 0−) of f(x) is equal to lim as x approaches 0 from the positive side of -f(x).

Conclude the relationship: Since lim as x → 0+ f(x) = 3, it follows that lim as x → 0− f(x) = -3, due to the odd function property.

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Odd Function

An odd function is a function f(x) that satisfies the condition f(-x) = -f(x) for all x in its domain. This symmetry about the origin implies that the graph of the function is symmetric with respect to the origin. Understanding this property is crucial for analyzing the behavior of the function as x approaches zero from both the positive and negative directions.

Limit from the Right

The limit of a function as x approaches a value from the right, denoted as limx→a+ f(x), describes the behavior of the function as x gets arbitrarily close to a from values greater than a. In this context, knowing limx→0+ f(x) = 3 indicates that as x approaches zero from the positive side, the function approaches the value 3.

Limit from the Left

The limit of a function as x approaches a value from the left, denoted as limx→a− f(x), describes the behavior of the function as x gets arbitrarily close to a from values less than a. For an odd function, the limit from the left can be deduced from the limit from the right due to the symmetry property, specifically limx→0− f(x) = -limx→0+ f(x).

Theory and ExamplesSuppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.