Squeeze theorem exercise example | Limits | Differential Calculus | Khan Academy | Summary and Q&A
TL;DR
The video explains how to create a compound inequality and uses the squeeze theorem to find the limit of a function.
Key Insights
- ☺️ Compound inequalities can be created to compare the values of multiple functions for specific x-values.
- 🔨 The squeeze theorem is a useful tool to find the limit of a function by comparing it to two other functions.
- ⛔ Using the definitions of functions and the squeeze theorem, it is possible to determine the value of the limit.
- 📈 The graph of the functions can provide visual confirmation of the conclusions drawn from the analysis.
- ☺️ The compound inequality f(x) ≤ g(x) ≤ h(x) holds true for the given x-values near 2.
- ⛔ The squeeze theorem allows us to make statements about the limits of functions based on the limits of other related functions.
- 📁 The squeeze theorem is a powerful tool in calculus that helps find limits when direct computation is difficult.
Transcript
The graphs of f of x, g of x, and h of x are shown below. Select and drag cards to create a compound inequality that orders the values of f of x, g of x, and h of x for x-values near 2 but not at 2 itself. So for any of the x-values that are depicted right over here, say, x is equal to 3, we see that h of 3 is the largest, f of 3 is the smallest, a... Read More
Questions & Answers
Q: How do you create a compound inequality that orders the values of three functions?
To create a compound inequality, compare the values of the three functions for the given x-values near 2. Use the graph to determine the order of the functions and write f(x) ≤ g(x) ≤ h(x).
Q: What are the definitions of f(x), g(x), and h(x)?
f(x) = 2x times the square root of x minus 1 minus 1, g(x) is a rational expression, and h(x) = e to the x minus 2.
Q: How is the squeeze theorem used in this analysis?
The squeeze theorem is used to find the limit as x approaches 2 of the three functions. By comparing the limits, it is determined that the value of the limit is equal to 1.
Q: Why does g(x) approach 1 based on the graph?
Since f(x) and h(x) both approach 1 as x approaches 2, g(x) must also approach 1. This is because g(x) is sandwiched between f(x) and h(x) for the given x-values.
Summary & Key Takeaways
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The video demonstrates how to create a compound inequality that orders the values of three functions for x-values near 2.
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The functions f(x), g(x), and h(x) are compared, and it is concluded that f(x) ≤ g(x) ≤ h(x) for the given x-values.
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The definitions of f(x), g(x), and h(x) are provided, and the squeeze theorem is used to find the limit as x approaches 2.