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# Limits

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### Limits

1. 1. Unit 1: Limits<br />Helga Hufflepuff<br />
2. 2. A limit is<br />
3. 3. A limit is the y-value of a graph as x approaches from both sides<br />
4. 4.
5. 5.
6. 6. A limit does not exist when the y-value is different as you approach the same x value from different sides<br />
7. 7. If the function is continuous…<br />you can draw the function without ever having to pick up your pen from the graph<br />The limit value is the same as the function value<br />
8. 8. Discontinuity<br />Types of Discontinuity:<br />Jump <br />Hole<br />Infinite (Asymptote)<br />
9. 9. Left- and Right-Hand Limits<br />Sometimes, limits do not approach the same y-value as they approach the same x-value, but the limit still exists<br />If approaching from the left -<br />If approaching from the right +<br />
10. 10. L’Hopital’s Rule<br />Used to calculate limits involving indeterminate forms; 0/0 or ∞/∞<br />Is there a limit?<br />
11. 11. 1) Let’s find the derivative of y/x:<br />2) Since x’=1, we can now find the limit.<br /> The limit is 1<br />
12. 12. The Squeeze Theorem<br />If the limit as h(x) approaches a is equal to the limit as f(x) approaches a—at point L-- and f(x)<g(x)<h(x), then g(x) equals L<br />
13. 13. Continuous VS. Differentiable<br />Differentiable– Does the derivative exist?<br />If a function is not continuous it cannot be differentiable on all reals<br />Continuous on all reals. Does the derivative exist<br />
14. 14. Differentiable?<br />If the function is continuous, we only need to worry about where derivative is undefined<br />Cusps <br />Corners<br />At x=2, the derivative is undefined<br />The function is differentiable on all reals except for where x=2<br />
15. 15. Continuous and<br /> Differentiable<br />
16. 16. In the Absolute Value function, the derivative is undefined at x=0<br />NOT DIFFERENTIABLE<br />
17. 17. Intermediate Value Theorem<br />If the function is continuous from [a,b], then there must be a point c in the interval [a,b] and it must have a y-value that is between f(a) and f(b) <br />
The text passage appears to be a series of slides or notes on the topic of limits in calculus, including definitions, examples, and theorems. It does not contain any direct mathematical questions or answers to such questions.
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Given a text passage, determine:
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    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

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- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# Limits

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### Limits

1. 1. Unit 1: Limits<br />Helga Hufflepuff<br />
2. 2. A limit is<br />
3. 3. A limit is the y-value of a graph as x approaches from both sides<br />
4. 4.
5. 5.
6. 6. A limit does not exist when the y-value is different as you approach the same x value from different sides<br />
7. 7. If the function is continuous…<br />you can draw the function without ever having to pick up your pen from the graph<br />The limit value is the same as the function value<br />
8. 8. Discontinuity<br />Types of Discontinuity:<br />Jump <br />Hole<br />Infinite (Asymptote)<br />
9. 9. Left- and Right-Hand Limits<br />Sometimes, limits do not approach the same y-value as they approach the same x-value, but the limit still exists<br />If approaching from the left -<br />If approaching from the right +<br />
10. 10. L’Hopital’s Rule<br />Used to calculate limits involving indeterminate forms; 0/0 or ∞/∞<br />Is there a limit?<br />
11. 11. 1) Let’s find the derivative of y/x:<br />2) Since x’=1, we can now find the limit.<br /> The limit is 1<br />
12. 12. The Squeeze Theorem<br />If the limit as h(x) approaches a is equal to the limit as f(x) approaches a—at point L-- and f(x)<g(x)<h(x), then g(x) equals L<br />
13. 13. Continuous VS. Differentiable<br />Differentiable– Does the derivative exist?<br />If a function is not continuous it cannot be differentiable on all reals<br />Continuous on all reals. Does the derivative exist<br />
14. 14. Differentiable?<br />If the function is continuous, we only need to worry about where derivative is undefined<br />Cusps <br />Corners<br />At x=2, the derivative is undefined<br />The function is differentiable on all reals except for where x=2<br />
15. 15. Continuous and<br /> Differentiable<br />
16. 16. In the Absolute Value function, the derivative is undefined at x=0<br />NOT DIFFERENTIABLE<br />
17. 17. Intermediate Value Theorem<br />If the function is continuous from [a,b], then there must be a point c in the interval [a,b] and it must have a y-value that is between f(a) and f(b) <br />
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The text passage appears to be a series of slides or notes on the topic of limits in calculus, including definitions, examples, and theorems. It does not contain any direct mathematical questions or answers to such questions.

TYPE: NO_QUESTION