Approximating limits using tables | Limits and continuity | AP Calculus AB | Khan Academy

TL;DR

Analyzing the limit of a function as x approaches three to determine its value.

Transcript

• This video we're going to try to get a sense of what the limit as x approaches three of x to the third minus three x squared over five x minus 15 is. Now when I say get a sense, we're gonna do that by seeing what values for this expression we get as x gets closer and closer to three. Now one thing that you might wanna try out is, well what happen... Read More

Key Insights

• 😑 The expression x to the third minus three x squared over five x minus 15 is not defined at x equals three as it results in an indeterminate form of zero over zero.
• 😑 To estimate the limit of the expression as x approaches three, two tables are set up: one for values less than three (from the left) and one for values larger than three (from the right).
• 😚 By calculating the expression for values closer and closer to three, an estimation is made that the limit approaches 1.8.
• ⛔ The limit exists if both the limit from the left and the limit from the right converge to the same value.
• 🗨️ The limit from the left refers to values less than three, approaching three from the left side.
• 🗯️ The limit from the right refers to values larger than three, approaching three from the right side.
• 📫 The estimation of the limit becomes more accurate as x values get closer and closer to three.

Questions & Answers

Q: What happens to the expression when x equals three?

When x equals three, the expression results in an indeterminate form of zero over zero, as both the numerator and denominator become zero.

Q: Why are two tables set up for approximation?

Two tables are set up to estimate the limit from both sides of three - values less than three (from the left) and values larger than three (from the right). This ensures that both approaches converge to the same value for the limit.

Q: Is the limit from the left negative values?

No, the limit from the left refers to values less than three, approaching three from the left side. It does not necessarily involve negative values, but rather values closer to three from the left side.

Q: How is the limit estimate calculated?

The limit estimate is calculated by plugging in x values closer and closer to three into the expression and observing the resulting values. Based on this, an estimation is made for the value that the limit approaches.

Summary & Key Takeaways

• The expression x to the third power minus three times x squared over five times x minus 15 is not defined when x equals three, as it results in an indeterminate form of zero over zero.

• To estimate the limit of the expression as x approaches three, two tables are set up: one for values less than three (from the left) and one for values larger than three (from the right).

• By calculating the expression for values closer to three, it is estimated that the limit approaches 1.8.