Chapter 1 Limits and Their Properties What is

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Chapter 1 Limits and Their Properties

Chapter 1 Limits and Their Properties

What is Calculus? Mathematics of change Pre Calculus Limit Process Calculus Differential Calculus vs.

What is Calculus? Mathematics of change Pre Calculus Limit Process Calculus Differential Calculus vs. Integral Calculus deals with the variation of a function with respect to changes in the independent variable deals with integration and its application in the solution of differential equations and in determining areas or volumes

Limits A function has the limit L as x approaches a *taking values of

Limits A function has the limit L as x approaches a *taking values of x that are really close to ‘a’, what is the function approaching? methods to evaluate limits: table, graph, algebra

Find the limit of X f(x) 1. 5 1. 9 Example : 1. 999

Find the limit of X f(x) 1. 5 1. 9 Example : 1. 999 2 ? as x approaches 2 2. 001 2. 5

Example : Evaluate *substituting 1 will give a 0 in the denominator error at

Example : Evaluate *substituting 1 will give a 0 in the denominator error at x = 1

Examples: Evaluate 1. 4. 2. 5. 3. 6.

Examples: Evaluate 1. 4. 2. 5. 3. 6.

More Limit Notation Approach a from the LEFT side Approach a from the RIGHT

More Limit Notation Approach a from the LEFT side Approach a from the RIGHT side Approach a from BOTH sides If , then

Evaluating Limits Graphically Examples: Does a limit exist at f(0)? Does a limit exist

Evaluating Limits Graphically Examples: Does a limit exist at f(0)? Does a limit exist as x approaches 3?

Example: Evaluate each limit Evaluating Limits Graphically For which interval(s) of c does the

Example: Evaluate each limit Evaluating Limits Graphically For which interval(s) of c does the limit of f(x) exist as x approaches c?

Example: Evaluate. Evaluating Limits Graphically For which interval(s) of c does the limit of

Example: Evaluate. Evaluating Limits Graphically For which interval(s) of c does the limit of f(x) exist as x approaches c?

Evaluating Limits Graphically Examples: Does a limit exist as x approaches ∞? Or -

Evaluating Limits Graphically Examples: Does a limit exist as x approaches ∞? Or - ∞?

Evaluating Limits Graphically Example: Does a limit exist as x approaches 0? For which

Evaluating Limits Graphically Example: Does a limit exist as x approaches 0? For which interval(s) of c does the limit of f(x) exist as x approaches c?

Evaluating Limits Graphically Examples: Does a limit exist as x approaches -∞ ? 2?

Evaluating Limits Graphically Examples: Does a limit exist as x approaches -∞ ? 2? ∞?

Evaluating Limits Graphically Example: For which interval(s) of c does the limit of f(x)

Evaluating Limits Graphically Example: For which interval(s) of c does the limit of f(x) exist as x approaches c?

Examples: Evaluate Evaluating Limits Graphically For which interval(s) of c does the limit of

Examples: Evaluate Evaluating Limits Graphically For which interval(s) of c does the limit of f(x) exist as x approaches c?

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c does the limit of f(x) exist as x approaches c?

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c does the limit of f(x) exist as x approaches c?

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c does the limit of f(x) exist as x approaches c?

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c

Evaluating Limits Graphically Example: Graph the piece-wise function and for which interval(s) of c does the limit of f(x) exist as x approaches c?

Finding Limits Algebraically For all functions, first try substitution! If you get an indeterminate

Finding Limits Algebraically For all functions, first try substitution! If you get an indeterminate form, you must try something else. (Table, Graph, or Algebra) Indeterminate forms:

Evaluating Limits Algebraically Examples: Short and sweet! 1. 2. 3.

Evaluating Limits Algebraically Examples: Short and sweet! 1. 2. 3.

Evaluating Limits Algebraically More Examples…Not so short and just a little sweet. 4. 5.

Evaluating Limits Algebraically More Examples…Not so short and just a little sweet. 4. 5. 6.

Evaluating Limits Algebraically More Examples…Not so short and not so sweet. 7.

Evaluating Limits Algebraically More Examples…Not so short and not so sweet. 7.

Evaluating Limits Algebraically More Examples…Oh boy…. 8. If , find

Evaluating Limits Algebraically More Examples…Oh boy…. 8. If , find

More Examples…Oh boy…. 9. If , find Evaluating Limits Algebraically

More Examples…Oh boy…. 9. If , find Evaluating Limits Algebraically

Infinite Limits evaluating limits when x approaches ±∞ Examples: 1. 2. 3. 4.

Infinite Limits evaluating limits when x approaches ±∞ Examples: 1. 2. 3. 4.

Infinite Limits evaluating limits when x approaches ±∞ Examples: 5. 6. 7. 8.

Infinite Limits evaluating limits when x approaches ±∞ Examples: 5. 6. 7. 8.

Infinite Limits – Shortcuts evaluating limits when x approaches ±∞

Infinite Limits – Shortcuts evaluating limits when x approaches ±∞

Infinite Limits Practice: Evaluate each limit. 1. 2. 3. 4. 5. 6. 7. 8.

Infinite Limits Practice: Evaluate each limit. 1. 2. 3. 4. 5. 6. 7. 8. Infinite Limits

Trigonometric Limits by Substitution 1. 2. 3.

Trigonometric Limits by Substitution 1. 2. 3.

Trigonometric Limits by Substitution 4. 5. 6.

Trigonometric Limits by Substitution 4. 5. 6.

Trigonometric Limits Provable by the Squeeze Theorem 1. 2.

Trigonometric Limits Provable by the Squeeze Theorem 1. 2.

Trigonometric Limits Provable by the Squeeze Theorem 3. 4. 5.

Trigonometric Limits Provable by the Squeeze Theorem 3. 4. 5.

LIMITS QUIZ 1

LIMITS QUIZ 1

Continuity A function f(x) is continuous if and only if: 1. 2. 3. Removable

Continuity A function f(x) is continuous if and only if: 1. 2. 3. Removable discontinuity vs. Nonremovable discontinuity

Continuity Example: Graph f(x). Is f(x) continuous? Justify using the 3 qualities of a

Continuity Example: Graph f(x). Is f(x) continuous? Justify using the 3 qualities of a continuous function.

Continuity Example: Is f(x) continuous at x = 4? Justify using the 3 qualities

Continuity Example: Is f(x) continuous at x = 4? Justify using the 3 qualities of a continuous function.

LIMITS AND CONTINUITY QUIZ 2

LIMITS AND CONTINUITY QUIZ 2

Continuity on a closed interval [a, b] f(x) is continuous on [a, b] if

Continuity on a closed interval [a, b] f(x) is continuous on [a, b] if it is continuous on the open interval (a, b) and AND Example: Determine the interval for which f(x) is continuous

Continuity Example: Graph f(x). Discuss any discontinuities and the interval(s) for which f(x) is

Continuity Example: Graph f(x). Discuss any discontinuities and the interval(s) for which f(x) is continuous.

Continuity Example: Discuss any discontinuities and the interval(s) for which f(x) is continuous.

Continuity Example: Discuss any discontinuities and the interval(s) for which f(x) is continuous.

Intermediate Value Theorem If f(x) is continuous on the closed interval [a, b] and

Intermediate Value Theorem If f(x) is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k.

Examples: a) Use the Intermediate Value Theorem to show has a zero in [0,

Examples: a) Use the Intermediate Value Theorem to show has a zero in [0, 1]. IVT b) Find c in [0, 3] such that f(c) = 11. Use the Intermediate Value Theorem to justify the answer.

LIMITS AND CONTINUITY TEST

LIMITS AND CONTINUITY TEST