Algebra Class: Limits and Piecewise Functions

Questions and Answers

What does x + x equal?

What is lim x->c [f(x) + f(x)] equal to?

2lim x->c f(x)

What is lim x->1 (f(-x) + g(x)/2) given lim x->-1 f(x) = 2 and lim x->1 g(x) = 6?

What is lim x->4 f(f(x)) if as x approaches 4, f(x) approaches 1?

lim x->1 f(x) Signup and view all the answers

When doing piecewise defined functions and limits, how do you know which equation to use?

Look at whether the limit is coming from the left or right. Signup and view all the answers

What should you do if the limit is coming from both sides?

Check limits from both sides. Signup and view all the answers

Study Notes

Algebraic Properties of Limits

The sum of a variable, x, added to itself is simplified to 2x.
The limit as x approaches c of the sum of a function f(x) and itself equals twice the limit of the function: lim x->c [f(x) + f(x)] = 2lim x->c f(x).

Evaluating Limits with Piecewise Functions

Example limits:
- lim x->-1 f(x) = 2,
- lim x->1 f(x) = 4,
- lim x->1 g(x) = 6.
To find lim x->1 (f(-x) + g(x)/2), substitute values under given limits:
- lim x->1 f(-x) uses lim x->1 f(x) = 2,
- g(1) = 6. Therefore, (2 + 6/2) = 5.

Nested Limits

For lim x->4 f(f(x)), if f(x) approaches 1 as x approaches 4, then substitute into the function:
- This results in lim x->4 f(f(1)) which simplifies to lim x->1 f(x). Evaluate f(x) at the new limit.

Piecewise Defined Functions

When working with piecewise defined functions and limits, identify which equation to use based on the direction of the limit:
- For left-hand limits (x->-5⁻), apply the condition x < -5. Use this equation as x is approaching from the left side of -5.

Evaluating Two-Sided Limits

Always check the limits from both sides for x->c.
If the left-hand limit matches the right-hand limit, that value is the limit; if they differ, the limit does not exist (DNE).

Description

This quiz focuses on the algebraic properties of limits and piecewise functions. It includes flashcards that explore various limit scenarios and algebraic expressions to enhance your understanding of these crucial concepts in calculus.