Limits of Functions

Identify the limit of the function f(x) = 2x - 1 as x approaches 3.

• The limit of the function f(x) = 2x - 1 as x approaches 3 is 5.

• The limit of the function f(x) = 2x - 1 as x approaches 3 is 0.

• The limit of the function f(x) = 2x - 1 as x approaches 3 is -3.

• The limit of the function f(x) = 2x - 1 as x approaches 3 is 7.

Evaluate the limit of the function g(x) = x^2 - 4x + 3 as x approaches 2 using a table of values.

• The limit of g(x) as x approaches 2 is 5.

• The limit of g(x) as x approaches 2 is -2.

• The limit of g(x) as x approaches 2 is 10.

• The limit of g(x) as x approaches 2 is 1.

Explain the concept of one-sided limits and provide an example.

• The one-sided limit of a function is always equal to the function value at that point

• One-sided limits only exist for odd-degree polynomial functions

• One-sided limits are only applicable to functions with a continuous domain

• For example, the one-sided limit of the function f(x) = 1/x as x approaches 0 from the positive side (right-hand limit) is +infinity, and as x approaches 0 from the negative side (left-hand limit) is -infinity.

Determine if the limit of the function h(x) = 1/x exists as x approaches 0.

• The limit is 0.

• The limit is 1.

• The limit is undefined.

• The limit does not exist.

Find the limit of the function k(x) = |x - 2| as x approaches 2 from the left.

• 2

• -2

• 1

• 0

Using a table of values, calculate the limit of the function m(x) = 3x^2 - 2x - 1 as x approaches 1.

• The limit of the function m(x) = 3x^2 - 2x - 1 as x approaches 1 is 0.

• The limit of the function m(x) = 3x^2 - 2x - 1 as x approaches 1 is 5.

• The limit of the function m(x) = 3x^2 - 2x - 1 as x approaches 1 is undefined.

• The limit of the function m(x) = 3x^2 - 2x - 1 as x approaches 1 is -1.

Discuss the conditions for a limit to exist and provide an example to illustrate.

• For example, the limit of the function f(x) = 2x as x approaches 3 exists because the left-hand limit and the right-hand limit both approach 6 as x gets closer to 3.

• The limit exists if the function is increasing at that point

• A limit exists if the function is defined at that point

• The limit exists if the function is continuous at the point

Evaluate the limit of the function n(x) = (x^2 - 1)/(x - 1) as x approaches 1.

• 3

• 0

• 2

• x

Explain the difference between a limit existing and a function being continuous at a point.

• The difference is that a limit existing only requires the function to approach a specific value, while continuity at a point requires the limit to exist and the function value to be equal to the limit.

• A limit exists when the function value is equal to the limit, while continuity at a point requires the limit to be undefined

• A limit exists when the function value is equal to the limit, while continuity at a point only requires the function to approach a specific value

• A limit exists when the function value is equal to the limit, while continuity at a point does not require the limit to exist

Calculate the limit of the function p(x) = sqrt(x + 1) - 1 as x approaches 0.

• The limit of p(x) as x approaches 0 is undefined

• The limit of p(x) as x approaches 0 is 0.

• The limit of p(x) as x approaches 0 is 1

• The limit of p(x) as x approaches 0 is -1