Understanding Limits in Rational Functions

The video tutorial explains how to determine the limit of a rational function as a approaches zero. It begins by discussing the indeterminate form of zero divided by zero and the need for algebraic simplification. The tutorial demonstrates how to simplify the function using positive exponents and complex fraction techniques, including finding the least common denominator. It then covers factoring and simplifying the expression to eliminate removable discontinuities. Finally, the video shows how to find the limit using direct substitution, resulting in a simplified expression.

What is the indeterminate form encountered when using direct substitution in this problem?

• Infinity over zero

• Zero over zero

• One over zero

• Zero over one

What is the purpose of using algebraic techniques in finding limits?

• To change the function completely

• To eliminate removable discontinuities

• To make the function more complex

• To avoid using limits

What is the equivalent expression for four plus a to the power of negative two?

• Four plus a squared

• One divided by four plus a squared

• One divided by four plus a

• Four plus a to the power of two

What is the LCD used to simplify the complex fraction in the problem?

• 16 times four plus a

• 16 times the square of four plus a

• Four plus a

• 16

What is the result of squaring four plus a?

• 16 plus 8a plus a squared

• 16 plus 4a plus a squared

• 8a plus a squared

• 16 plus a squared

What common factor is factored out from the numerator?

• Positive one

• Negative one

• Positive a

• Negative a

What is the simplified form of the rational expression after factoring?

• The opposite of a plus eight divided by 16 times the square of four plus a

• The opposite of a plus eight divided by four plus a

• A plus eight divided by four plus a

• A plus eight divided by 16 times the square of four plus a

What is the final limit of the function as a approaches zero?

• 1/16

• Negative 1/16

• Negative 1/32

• 1/32

Why was direct substitution not possible initially?

• The limit was infinite

• There was a removable discontinuity

• The function was too complex

• The function was undefined

What does simplifying out the common factor of a achieve?

• It changes the function

• It eliminates the removable discontinuity

• It makes the function more complex

• It avoids the use of limits