Calculus Limits and Continuity

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Questions and Answers

What is necessary for a function to be continuous at a point x = e?

f(e) is defined and equal to lim f(x). (correct)

f(e) must be positive.

lim f(x) must be greater than zero.

lim f(x) must approach infinity.

What characterizes a removable discontinuity?

Both one-sided limits are equal.

The limit exists but is not equal to the function value or f(e) is undefined. (correct)

The limit does not exist at that point.

The limit exists and equals the function value.

How can a removable discontinuity be resolved?

By changing the limit at the discontinuous point.

By redefining the function at the discontinuity to match the limit. (correct)

By ensuring one-sided limits are equal.

By increasing the degree of the polynomial.

In a jump discontinuity, what relationship do the one-sided limits have?

They exist but are not equal. Signup and view all the answers

What is a removable singularity?

A point where the function is not defined but the limit exists. Signup and view all the answers

If g(x) = (x^3 - 8)/(x - 2) has a removable discontinuity at x = 2, how should g(2) be defined for continuity?

g(2) = 12 Signup and view all the answers

What happens when a function has a removable discontinuity?

The function can be made continuous by defining its value appropriately. Signup and view all the answers

Which of the following describes a function with a jump discontinuity?

One-sided limits exist but differ from one another. Signup and view all the answers

Under what condition is a function considered continuous at a point?

If both the limit exists and is equal to the function value at that point. Signup and view all the answers

What is a removable discontinuity?

A point where the left-hand limit and right-hand limit are equal but do not equal the function value. Signup and view all the answers

Which of the following functions is continuous at all points?

f(x) = sin(x) Signup and view all the answers

What does it indicate if the left-hand limit and right-hand limit of a function at a point are not equal?

There is a discontinuity at that point. Signup and view all the answers

Which of the following scenarios describes a function being discontinuous?

The left-hand limit and the right-hand limits are equal but differ from the function value. Signup and view all the answers

For which of the following values of x is the function f(x) = k continuous?

At all points in its domain for any constant value k. Signup and view all the answers

What is the implication of a function having limits that do not exist at a certain point?

The function has a discontinuity at that point. Signup and view all the answers

In a graph representing a continuous function, what should you observe?

The graph is a single piece with no interruptions. Signup and view all the answers

Study Notes

Limits and Continuity

Continuity in calculus indicates a function's graph has no breaks, akin to a single piece of wire.
Physical phenomena, such as position, velocity, and barometric pressure, are typically continuous with respect to time or altitude.
Functions must be continuous for proper mathematical analysis when determining rates of change.

Definition of Continuity

A function f is continuous at a point x = c if:
- f(c) is defined.
- The limit lim f(x) as x approaches c exists.
- The limit equals the function value, i.e., lim f(x) = f(c) as x approaches c.
If any of these conditions are not satisfied, the function is considered discontinuous.

Types of Functions and Continuity

Functions like f(x) = k (where k is a constant) and g(x) = x^n (where n is a whole number) are examples of continuous functions.
The limit of a constant function is equal to the constant itself. For polynomial functions, limits can generally be evaluated as their direct values.

Discontinuities

Discontinuities occur when:
- A limit exists but does not equal the function value.
- The function value is undefined.
Removable discontinuities can be corrected by redefining or assigning a specific value to the function at that point (e.g., changing f(2) to equal the limit).

Types of Discontinuities

Removable Discontinuity: Exists when lim f(x) exists but does not equal f(c) or f(c) is undefined. Can often be resolved by redefining f(c).
Jump Discontinuity: Occurs when the one-sided limits exist but are not equal, leading to a failure of continuity at the point without a straightforward fix.

Example of Removable Discontinuity

For the function g(x) = (x^3 - 8)/(x - 2):
- g(2) is undefined due to division by zero.
- The limit as x approaches 2 is found using the factorization method; for this function, lim g(x) = 12.
- Redefining g(2) = 12 allows for continuity at x = 2.

Visual Representation

Graphs can illustrate continuity and types of discontinuity effectively, highlighting clear breaks or jumps in the function values relative to the expected limits.

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Description

Explore the concept of limits and continuity in calculus with this quiz. Understand how functions behave without breaks or interruptions and relate these mathematical ideas to real-world phenomena. Assess your knowledge of continuous functions and their graphical representations.