Even function limits Suppose f is an even function where lim x→1^− f(x)=5 and lim x→1^+ f(x)=6. Find lim x→−1^− f(x) and limx→−1^+ f(x).

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Recall that an even function satisfies the property f(x) = f(-x) for all x in its domain.

Given that \( \lim_{x \to 1^-} f(x) = 5 \) and \( \lim_{x \to 1^+} f(x) = 6 \), use the even function property to relate these limits to the limits as x approaches -1.

For \( \lim_{x \to -1^-} f(x) \), consider that as x approaches -1 from the left, -x approaches 1 from the right. Therefore, \( \lim_{x \to -1^-} f(x) = \lim_{x \to 1^+} f(x) = 6 \).

For \( \lim_{x \to -1^+} f(x) \), consider that as x approaches -1 from the right, -x approaches 1 from the left. Therefore, \( \lim_{x \to -1^+} f(x) = \lim_{x \to 1^-} f(x) = 5 \).

Conclude that \( \lim_{x \to -1^-} f(x) = 6 \) and \( \lim_{x \to -1^+} f(x) = 5 \) using the properties of even functions.

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Even Functions

An even function is defined by the property f(x) = f(-x) for all x in its domain. This symmetry about the y-axis means that the function's values at positive and negative inputs are identical. Understanding this property is crucial for analyzing limits of even functions, as it allows us to relate the behavior of the function at positive and negative values.

Limits from the Left and Right

Limits from the left (denoted as lim x→c^− f(x)) and from the right (lim x→c^+ f(x)) describe the behavior of a function as it approaches a specific point c from either side. These one-sided limits are essential for determining the overall limit at that point, especially when the left and right limits differ, indicating a potential discontinuity.

Limit Properties of Even Functions

For even functions, the limits at negative inputs can be directly inferred from the limits at their positive counterparts. Specifically, if lim x→1^− f(x) = 5 and lim x→1^+ f(x) = 6, then by the even function property, we can conclude that lim x→−1^− f(x) = 6 and lim x→−1^+ f(x) = 5, reflecting the symmetry of the function around the y-axis.