1.4 continuity + review

in what value of x does the following function is discontinuous?

• 3

• -3

• 6

• -6

Which of the following best describes the continuity at x = 5?

• Continuous

• Removable Point Discontinuity

• Infinite Discontinuity

• Jump Discontinuity

Answer explanation

At x = 5, if the function is defined but has a hole (a point where it is not continuous), it indicates a removable point discontinuity. This means the limit exists, but the function value at that point is either missing or different.

Which of the following best describes the continuity at x = 5?

• Continuous

• Removable Point Discontinuity

• Infinite Discontinuity

• Jump Discontinuity

Answer explanation

At x = 5, if the function approaches infinity or negative infinity, it indicates an infinite discontinuity. This means the function does not have a finite limit at that point, making 'Infinite Discontinuity' the correct choice.

Which of the following best describes the continuity at x = 5?

• Continuous

• Removable Point Discontinuity

• Infinite Discontinuity

• Jump Discontinuity

Answer explanation

The function is continuous at x = 5 if the limit as x approaches 5 equals the function's value at that point. Since the answer is 'Continuous', it indicates no breaks or jumps in the function at x = 5.

• is a continuous function for all x-values

Answer explanation

f(x) has a removable discontinuity at x=2 because it can be simplified to f(x) = x + 2 for x ≠ 2. It is continuous everywhere except at x=2, but technically it is continuous at all points in its domain.

• Discontinuous at x=5; Jump

• Discontinuous at x=5; Removable            

• Continuous

• Discontinuous at x=5; Infinite

Answer explanation

The function f(x)=1/(x-5) is undefined at x=5, leading to a vertical asymptote. This indicates it is discontinuous at x=5 and the type of discontinuity is infinite, as the function approaches infinity.

• Discontinuous at x=3; Jump                 

• Discontinuous at x=3; Removable        

• Discontinuous at x=3; Infinite

• Continuous

Answer explanation

The function f(x) = (x^2 - 9)/(x - 3) is discontinuous at x=3 due to division by zero. However, it can be simplified to f(x) = x + 3 for x ≠ 3, indicating a removable discontinuity at x=3.

Examine the given function above and classify it accordingly.

• Continuous

• Discontinuous at x=-3; Jump

• Discontinuous at x=3; Jump

• Undefined

Answer explanation

The function is discontinuous at x=3 due to a jump, meaning the left-hand limit and right-hand limit at this point do not equal the function value. Thus, the correct classification is 'Discontinuous at x=3; Jump'.

• t=0, hole;
  t=3, t=8 vertical asymptotes

• t=0, vertical asymptote;
  t=3, t=8, holes

• t=0, hole
  t = -1, vertical asymptote. 

• t=-8, hole; 
  t=0, t = -3, vertical asymptotes. 

Answer explanation

The function h(t) has a hole at t=0 because the numerator and denominator both equal zero there. The vertical asymptotes occur at t=3 and t=8, where the denominator is zero but the numerator is not. Thus, the correct classification is t=0, hole; t=3, t=8 vertical asymptotes.

In f(x) = x² + 5x + 6 ∕  x + 3, at what value of x it will be a removable discontinuity?

• -3

• -2

• 2

• 3

Answer explanation

In f(x) = (x^2 + 5x + 6) / (x + 3), the numerator factors to (x + 2)(x + 3). The term (x + 3) cancels, indicating a removable discontinuity at x = -3.