Question

T/F: If \(\lim _{x \rightarrow 1^{-}} f(x)=5,\) then \(\lim _{x \rightarrow 1^{+}} f(x)=5\)

Short answer

False, the left and right limits can be different.

Step-by-step solution

Understanding One-Sided Limits

The problem involves two one-sided limits: the left-hand limit (denoted by \(\lim_{x \to 1^-} f(x)\)) and the right-hand limit (denoted by \(\lim_{x \to 1^+} f(x)\)). The left-hand limit is the value that \(f(x)\) approaches as \(x\) approaches 1 from the left, while the right-hand limit is the value that \(f(x)\) approaches as \(x\) approaches 1 from the right.

Defining the Statement to be True or False

The statement claims that if the left-hand limit is equal to 5, then the right-hand limit must also equal 5. In other words, it asserts that the limit as \(x\) approaches 1 from both sides has to be the same. Remember that the overall limit \(\lim_{x \to 1} f(x)\) exists only if both one-sided limits are equal and finite.

Considering a Counterexample

To determine if the statement is always true, consider a function that behaves differently as \(x\) approaches 1 from different sides. For instance, let \(f(x) = 5\) for \(x < 1\) and \(f(x) = 10\) for \(x > 1\). In this example, \(\lim_{x \to 1^-} f(x) = 5\) and \(\lim_{x \to 1^+} f(x) = 10\), which shows that the right-hand limit does not necessarily equal 5.

Final Conclusion

The statement is false. As demonstrated by the counterexample, it's possible for \(\lim_{x \to 1^-} f(x) = 5\) without \(\lim_{x \to 1^+} f(x)\) being equal to 5. For the overall limit to exist and equal to 5, both one-sided limits must be equal, which is not guaranteed from the condition given.

Key concepts

Left-Hand Limit

When we talk about the left-hand limit, we are considering what happens to the function as the variable approaches a specific value from the left side. In mathematical terms, for the limit as \(x\) approaches 1 from the left, which is written as \( \lim_{x \to 1^-} f(x) \), we look at the values of \(f(x)\) when \(x\) is slightly less than 1.

The left-hand limit tells us about the behavior of the function right up to the number, but strictly from the lower side.

• It involves understanding how \(f(x)\) behaves for values like 0.9, 0.99, 0.999 when examining the limit as \(x\) approaches 1 from the left.
• If \(f(x)\) approaches a particular number, then that number is the left-hand limit at \(x = 1\).

Grasping the concept of a left-hand limit helps in understanding how close \(f(x)\) gets to a number from one specific direction.

Right-Hand Limit

Similarly to the left-hand side, the right-hand limit involves what happens to \(f(x)\) as \(x\) approaches a specific value from the right. This is denoted as \( \lim_{x \to 1^+} f(x) \).

Here, we are concerned with the values of \(f(x)\) when \(x\) is just a little bit greater than 1, such as 1.1, 1.01, or 1.001.

• The right-hand limit reveals what \(f(x)\) trends toward as it gets closer to 1 from the right.
• It provides a clear view of \(f(x)\)'s tendencies from a slightly higher perspective to that specific point.

Recognizing the importance of right-hand limits is crucial for understanding how a function behaves as we near a point from above, providing symmetry to our analysis with the left-hand limit.

Counterexample

A counterexample can effectively demonstrate the incorrectness of an assumption in mathematics.

In the context of this exercise, we want to show that just because the left-hand limit equals a certain value, it doesn't automatically mean the right-hand limit matches it.

• Consider a function \(f(x)\) defined differently on either side of a point.
• A classic counterexample has \(f(x) = 5\) when \(x < 1\) and \(f(x) = 10\) when \(x > 1\).

For this function:

• As \(x\) approaches 1 from the left, \( \lim_{x \to 1^-} f(x) = 5\).
• Yet approaching from the right gives \( \lim_{x \to 1^+} f(x) = 10\).

This counterexample clearly shows that the values of the one-sided limits can differ, debunking the idea that they must be equivalent.

Existence of Limits

A crucial part of analyzing limits is determining when they exist. A limit at a particular point exists when the expected value is consistent, regardless of the direction from which we approach that point.

• If \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \), then \( \lim_{x \to a} f(x) \) exists and equals this common value.
• Conversely, if the left-hand and right-hand limits do not match, the overall limit at that point does not exist.

This concept is essential for continuity and overall analysis of functions.
Understanding the existence of limits helps in identifying any discontinuities and informs us whether the function has potential jumps or breaks at certain points. It's a foundational idea that supports many areas of mathematical study related to calculus and analysis.