Limits from tables for oscillating functions

TL;DR

When a function exhibits oscillations as it approaches a certain value, the limit may not exist. However, if the oscillations decrease in magnitude, it is reasonable to estimate that the function is approaching that value.

Transcript

• [Instructor] The function h is defined over the real numbers. This table gives a few values of h, so they give us for different x values, What is the value of h of x? What is a reasonable estimate for the limit of h of x as x approaches one? So with the table we can estimate the limit. We won't know 100% for sure, but it's a good way to estimate ... Read More

Key Insights

• ⛩️ Observing oscillations in a function as it approaches a certain value can provide hints about the existence of a limit.
• 😚 Oscillations becoming more wild as we get closer to a value often indicates that the limit does not exist.
• ⛔ If the magnitude of oscillations decreases as we approach a value, it is reasonable to estimate that the limit exists.
• ⛔ The behavior of a function as it approaches a value can be different from the value that the function is defined at, emphasizing the purpose of studying limits.

Questions & Answers

Q: How can we estimate the limit of a function using a table of values?

By analyzing the behavior of the function as we approach a specific value, we can make an estimate of the limit. This involves observing how the function changes for different x values that are getting closer to the target value.

Q: What is the significance of oscillations in determining the existence of a limit?

Oscillations alone do not indicate that a limit does not exist. However, if the oscillations become more significant or wilder as we approach a value, it is likely that the limit does not exist. On the other hand, if the oscillations decrease in magnitude, it is reasonable to estimate that the limit exists.

Q: Can the limit of a function be approached even if the function is defined at a different value?

Yes, the concept of limits allows us to study what a function approaches, even if the function is defined at a different value. In fact, the limit can approach a value that is different from the value at which the function is defined, as seen in the example where the limit is zero while the function is defined as four.

Q: Is it always possible to mathematically prove the existence or non-existence of a limit?

In many cases, it is not possible to mathematically prove the existence or non-existence of a limit. Estimating the limit based on the behavior of the function is often the best approach.

Summary & Key Takeaways

• The first example shows oscillations that become more significant on both sides as x approaches one, indicating that the limit does not exist.

• In the second example, the oscillations become less wild on both sides as x approaches three, suggesting that the limit is likely zero.

• The table highlights the importance of considering the behavior of a function as it approaches a value in order to estimate the limit.