Calculus: Limits of Functions

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Questions and Answers

What is the limit of a function at a point if both one-sided limits are equal?

The limit can only be calculated if the point is not a breaking point.

The limit does not exist.

The limit is infinite.

The limit is the value of the one-sided limits. (correct)

Which of the following represents an indeterminate limit?

3/0

∞ + 2

10/0

5 - 5 (correct)

When determining the limit of a function as x approaches positive or negative infinity, what can the result be?

Only positive or negative infinity

A real number, positive infinity, negative infinity, or the limit may not exist (correct)

Only a real number

Only the limit exists if the function is continuous

What indicates that the limit of a function at a given point does not exist?

The left and right limits are different. Signup and view all the answers

Which of the following situations does not lead to an indeterminate form during limit calculation?

3/5 Signup and view all the answers

What does the notation $ ext{lim}_{x o a^{-}} f(x)$ represent?

The limit of the function approached from the left side of a point. Signup and view all the answers

When do equal lateral limits imply that a function has a limit at a point?

If both lateral limits exist and are equal. Signup and view all the answers

What is true about the limits $ ext{lim}{x o 2^{-}} f(x) = 3$ and $ ext{lim}{x o 2^{+}} f(x) = 6$?

The overall limit at x = 2 does not exist. Signup and view all the answers

In which scenario can a function be continuous at a point a?

When both lateral limits exist and are equal, and the function value f(a) equals that limit. Signup and view all the answers

What concept does the term 'limit' refer to in the context of a function?

The value the function approaches as x gets closer to a specific point. Signup and view all the answers

Which of the following represents an indeterminate form?

$rac{ ext{infinity}}{ ext{infinity}}$ Signup and view all the answers

What is the indeterminate form that results from zero raised to the power of zero?

$0^0$ Signup and view all the answers

Which of the following combinations describes zero times infinity?

$0 imes ext{infinity}$ Signup and view all the answers

Which of the following indicates an indeterminate form related to infinite expressions?

$ ext{infinity}^0$ Signup and view all the answers

Which pair of forms is considered indeterminate?

$ ext{infinity} - ext{infinity}$, $rac{0}{0}$ Signup and view all the answers

Study Notes

Limits and Limits of Functions

A limit is the value a function approaches as the input approaches a certain value.
The limit of a function at a point is the value the function approaches as x gets closer to that point.
Left-hand limit: The limit of a function from the left is denoted by $\lim_{x \to a^{-}} f(x)$.
Right-hand limit: The limit of a function from the right is denoted by $\lim_{x \to a^{+}} f(x)$.
If the left-hand and right-hand limits are equal, the limit of the function exists at that point.
If the left-hand and right-hand limits are not equal, the limit of the function does not exist at that point.

Limits of Functions Defined by Parts

If a function is defined in pieces, you need to consider the limits from both the left and right at the points where the function changes.
If both one-sided limits are equal, the limit of the function at that point exists and is equal to that value.

Limits at Infinity

The limit of a function as x tends to positive or negative infinity examines the function's behavior as x gets very large.
The limit at infinity could be a real number, positive infinity, negative infinity, or may not exist.

Indeterminate Limits

Indeterminate limits are mathematical expressions that appear in limit calculations, where the result is not defined.
Some common indeterminate forms are:
- Infinity minus infinity (∞ − ∞)
- Number divided by zero (k/0)
- Zero divided by zero (0/0)
Indeterminate forms can be resolved using various techniques, including algebraic manipulation and L'Hopital's Rule.

Indeterminate Forms in Notes

The notes discuss various indeterminate forms, including:
- Infinite over infinite: $\frac{\infty}{\infty}$
- Zero to the power of zero: $0^0$
- Zero times infinity: $0 \times \infty$
- Infinity to the power of zero: $\infty^0$
- Zero raised to the power of zero: $0^0$
The notes provide examples and descriptions of each of these forms, explaining their behavior and how they can be handled in limit calculations.

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Description

This quiz covers the concept of limits and their application in functions, including left-hand and right-hand limits, as well as limits of piecewise functions. You'll explore how to determine if a limit exists and the behavior of functions as they approach specific points or infinity. Test your understanding of these fundamental calculus concepts!