Understanding Discontinuities in Rational Functions

The video tutorial explains how to identify discontinuities in a rational function. It discusses why setting the denominator to zero doesn't work in this case and highlights the importance of division by zero. The tutorial identifies a discontinuity at x equals two and classifies it as non-removable. It further distinguishes between jump and infinite discontinuities, using graph analysis to confirm a jump discontinuity at x equals two.

What is the initial step suggested to find the x values where the function is discontinuous?

• Evaluate the function at x = 0.

• Find the derivative of the function.

• Set the entire denominator equal to zero.

• Set the numerator equal to zero.

Why does the initial approach of setting the denominator to zero fail?

• Because the function is not defined for any x.

• Because e raised to any power is always negative.

• Because e raised to any power is always non-negative.

• Because the numerator is zero.

What condition must be met to find the discontinuity in the function?

• The exponent must cause division by zero.

• The denominator must be zero.

• The exponent must be zero.

• The numerator must be zero.

At which x value does the function have a discontinuity?

• x = 2

• x = 0

• x = -2

• x = 4

What type of discontinuity occurs at x = 2?

• Continuous

• Non-removable

• Oscillating

• Removable

What distinguishes a non-removable discontinuity?

• It occurs at zeros of both numerator and denominator.

• It occurs at zeros of the denominator not in the numerator.

• It occurs at zeros of the numerator not in the denominator.

• It occurs when the function is continuous.

What is a characteristic of jump discontinuity?

• The graph has a vertical asymptote.

• The graph is continuous.

• The graph oscillates.

• The graph has a vertical break.

What happens to the one-sided limits at a jump discontinuity?

• They do not exist.

• They are equal.

• They exist but are not equal.

• They are infinite.

As x approaches 2 from the left, what does the function value approach?

• Negative two

• Infinity

• Positive two

• Zero

As x approaches 2 from the right, what does the function value approach?

• Negative two

• Infinity

• Positive two

• Zero