Understanding Limits and Algebraic Simplification

The video tutorial explains how to find the limit of a function as x approaches 1. Initially, direct substitution results in an indeterminate form, prompting the need for simplification. The denominator is factored using the difference of squares, and algebraic long division is applied to the numerator. This simplification allows the cancellation of terms, leading to a straightforward calculation of the limit, which is found to be 3/2.

What is the initial problem encountered when trying to find the limit as x approaches 1 for the given expression?

• The expression simplifies to a constant.

• The expression is undefined for all x.

• The expression has no limit.

• The expression results in an indeterminate form 0/0.

What mathematical property is used to simplify the denominator of the expression?

• Sum of cubes

• Binomial theorem

• Difference of squares

• Quadratic formula

What technique is used to factor the numerator of the expression?

• Synthetic division

• Partial fraction decomposition

• Algebraic long division

• Completing the square

After factoring, what expression is obtained for the numerator?

• x - 1 times x squared minus x minus 1

• x - 1 times x squared plus x minus 1

• x - 1 times x squared plus x plus 1

• x - 1 times x squared minus x plus 1

What is the simplified expression for the limit as x approaches 1?

• x squared plus x plus 1 over x plus 1

• x squared minus x minus 1 over x plus 1

• x squared minus x plus 1 over x plus 1

• x squared plus x minus 1 over x minus 1

What is the final result of the limit as x approaches 1?

• 2/3

• 3/2

• 1/3

• 1/2