Evaluating Limits and Rationalization Techniques

The video tutorial explains how to evaluate limits analytically without relying on graphs or tables. It covers basic examples, such as limits as x approaches 3 and pi, and introduces methods like factoring and rationalizing to solve more complex limit problems. The tutorial also discusses the importance of understanding limits graphically, highlighting how graphs can have holes that affect limit evaluation. By the end, viewers should understand how to manipulate expressions to find limits and interpret them graphically.

What is the first step in evaluating limits analytically?

• Graph the function

• Directly substitute the limit value

• Use a table of values

• Differentiate the function

When can you directly substitute the limit value into a function?

• When the function is undefined

• When the function has a hole

• When the function is continuous at that point

• When the function is discontinuous

What does a 0/0 form indicate when evaluating limits?

• The limit is infinite

• Algebraic manipulation is needed

• The function is continuous

• The limit does not exist

In the example of factoring, what was the problematic term that caused the indeterminate form?

• x + 3

• x minus 6

• x squared

• x - 2

What is the result of factoring the expression x^2 - 2x over x^2 - 4?

• x over x + 2

• x - 2 over x + 2

• x + 2 over x - 2

• x over x - 2

What technique is used to evaluate limits involving square roots?

• Graphing

• Rationalizing

• Factoring

• Using a table

What is the conjugate of the expression sqrt(x + 1) - 1?

• 1 - sqrt(x + 1)

• x + 1

• sqrt(x + 1) - 1

• sqrt(x + 1) + 1

After rationalizing the expression sqrt(x + 1) - 1 over x, what is the simplified form?

• sqrt(x + 1) over x

• 1 over sqrt(x + 1) + 1

• 1 over x

• x over sqrt(x + 1) - 1

What happens to the x terms in the expression x over x times sqrt(x + 1) + 1 after simplification?

• They remain unchanged

• They add up

• They cancel out

• They multiply

What is the final limit of the expression sqrt(x + 1) - 1 over x as x approaches 0?

• 2

• 0

• 1

• 1/2