Calculus Unit 2: Limit of Function

Questions and Answers

Which interval notation represents the number line with a closed circle at 4 and an open circle at 7?

[4, 7) (correct)

[4, 7]

(4, 7]

(4, 7)

What does x tend to when examining the limit where x approaches 0 in the function y = (x² - 4) / (x - 2)?

2 (correct)

-2

Which of the following explains the limit behavior of the function f(x) = (3x + 2) / (x + 1) as x approaches infinity?

The limit approaches 2.

The limit approaches 3. (correct)

The limit approaches 0.

The limit is undefined.

What is the interval notation for the graph that shows an open circle at -5 and an open circle at 3?

(-5, 3) Signup and view all the answers

In the context of functions, what does the phrase 'x tends to infinity' signify?

The function grows without bound. Signup and view all the answers

Which notation represents an interval that includes neither endpoints?

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Which of the following correctly describes an interval that includes both endpoints?

[a, b] Signup and view all the answers

In which interval notation are endpoints excluded?

(x, y) Signup and view all the answers

What is the interval notation for all numbers greater than or equal to -2?

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What is the notation for an interval that includes only the endpoint -2 and not 3?

[ -2, 3 ) Signup and view all the answers

Which of the following correctly represents the set of all numbers less than 4?

( -∞, 4 ] Signup and view all the answers

An open interval on the number line excludes which of the following?

Both endpoints Signup and view all the answers

Given the interval notation [2, 5), which of the following is true?

2 is included, but 5 is not. Signup and view all the answers

Which statement accurately describes the interval notation (-2, 3)?

Includes neither -2 nor 3. Signup and view all the answers

Study Notes

Limit of Function

Calculus addresses dynamic situations, determining rates of change like economic growth or speed.
The concept of a limit is fundamental to the development of calculus.
Real numbers can be represented as points on a number line, allowing visualization of algebraic expressions and intervals.

Real Numbers and Intervals

A number line visually represents various types of real numbers, such as integers, fractions, and irrational numbers.
Intervals are defined as sets of all real numbers between two points, indicated using special notation:
- Open interval: (a, b) does not include endpoints a or b.
- Closed interval: [a, b] includes both a and b.
- Half-open interval: [a, b) includes a but not b, or (a, b] includes b but not a.
Example intervals on a number line include [-2, ∞) and (-∞, -1).

Limit Concepts

Terminology:
- x tends to zero (x → 0): The function approaches a specific number as x gets close to 0.
- x tends to a (x → a): The function approaches a certain value as x approaches any real number a.
- x tends to infinity (x → ∞): Function values behave as x grows indefinitely.
The limit of a function as x approaches some c is defined through the ε-δ criterion, ensuring that f(x) can be made arbitrarily close to L by choosing x sufficiently close to c.

Limit of a Sequence

A sequence {Sn} converges to a limit L if for every ε > 0, there exists an n such that |Sn - L| < ε for all n ≥ 1.
The notation lim {Sn} = L signifies this limit convergence.

Example Sequences

Example sequence Sn = (n + 2)/n^2 converges to 0 as n increases, observable through graphical representation.
Additional sequences can be examined to establish limits and behaviors in relation to their terms.

Theorems on Limits

Sum of Limits:
- If f(x) and g(x) approach limits L and M respectively, then lim[f(x) + g(x)] = L + M.
Difference of Limits:
- Similarly, lim[f(x) - g(x)] = L - M.

Example Applications

Practical examples illustrate how to calculate limits of functions using algebraic techniques, confirming theorems like sum, difference, product, and quotient of limits.
Provides logical steps demonstrating how to apply these concepts to specific functions across various computations.

Important Facts

Special case rules for calculating limits include the linearity rule, product rule, quotient rule, and root rule; all help derive limits systematically.
The approaches used in calculating limits facilitate deeper understanding of function behavior near defined values as well as at infinity.

Summary

Mastery of limits is essential for understanding calculus, allowing students to tackle real-world dynamic problems through mathematical analysis and function behavior.

Description

Explore the fundamental concepts of limits in calculus with this quiz. Understand how to analyze dynamic situations and apply these techniques to real-world problems. Test your knowledge on the behavior of functions as they approach specific values.