1.13 Removing Discontinuities

Cards (49)

  • In a jump discontinuity, the function approaches different limits from the left and right at a specific point.
  • Give an example of a function with a removable discontinuity.
    f(x) = \frac{x^2 - 4}{x - 2}</latex>
  • Steps to remove a removable discontinuity:
    1️⃣ Factor the numerator and denominator
    2️⃣ Simplify the expression
    3️⃣ Check the limit as x approaches the discontinuity
    4️⃣ Replace the discontinuous function with a continuous equivalent
  • A removable discontinuity occurs when the function has a hole or jump at a point, but the limit exists.
  • By canceling common factors, removable discontinuities can be eliminated.

    True
  • A removable discontinuity occurs when the function has a hole or jump
  • What is the first step when removing a removable discontinuity by factoring the numerator and denominator?
    Factor both numerator and denominator
  • When simplifying \frac{x^{2} - 9}{x^{2} +6x+ 6x + 9}, the denominator factors to (x + 3)(x + 3)
  • The limit of x2−9x−3\frac{x^{2} - 9}{x - 3} as xx approaches 3 is 6
  • What is the domain restriction needed for the function f(x)=f(x) =1x \frac{1}{x}?

    x≠0x \neq 0
  • What feature of a function indicates an infinite discontinuity?
    Vertical asymptote
  • Match the type of discontinuity with its definition:
    Removable ↔️ Limit exists at the point
    Jump ↔️ Sudden change in value
    Infinite ↔️ Vertical asymptote
  • Factoring the numerator and denominator is essential for removing removable discontinuities.
  • What is the simplified form of \( f(x) = \frac{x^2 - 9}{x - 3} \) after canceling common factors?
    x + 3
  • The left-hand and right-hand limits must be equal for a limit to exist at a discontinuity.
    True
  • Steps to replace a discontinuous function with a continuous equivalent:
    1️⃣ Identify the discontinuity
    2️⃣ Calculate the limit
    3️⃣ Redefine the function
  • Discontinuities occur in functions where the graph breaks or skips a point
  • An infinite discontinuity occurs when the function approaches infinity as it nears a specific point.
    True
  • The function f(x)=f(x) =1/x 1 / x has an infinite discontinuity at x=x =0 0.

    True
  • How is an infinite discontinuity removed?
    Restrict the domain
  • What is the example function given for an infinite discontinuity?
    f(x)=f(x) =1x \frac{1}{x}
  • What is the simplified form of the function f(x)=f(x) =x2−9x−3 \frac{x^{2} - 9}{x - 3} after removing the discontinuity?

    f(x)=f(x) =x+ x +3 3
  • A jump discontinuity exists when the limit exists at the point of discontinuity.
    False
  • Steps to remove a removable discontinuity by factoring the numerator and denominator:
    1️⃣ Factor both the numerator and denominator
    2️⃣ Cancel common factors
    3️⃣ Redefine the function at the point of discontinuity
  • Why is it important to check the limit as x</latex> approaches a discontinuity?
    To determine if it is removable
  • A jump discontinuity can be removed by redefining the function at the point of discontinuity.
    False
  • What type of discontinuity occurs when a function has a hole at a point but the limit exists?
    Removable
  • Identifying and removing discontinuities is a crucial step in analyzing the behavior of a function.
    True
  • What is an example of a function with an infinite discontinuity?
    f(x) = 1/x
  • What is one benefit of factoring when dealing with discontinuities?
    Simplifies complex expressions
  • To simplify an expression, identify common factors and cancel them.
  • What is the limit of \( \frac{x^2 - 4}{x - 2} \) as \( x \to 2 \)?
    4
  • What is the limit of \( \frac{x^2 - 9}{x - 3} \) as \( x \to 3 \)?
    6
  • What is a removable discontinuity characterized by?
    Limit exists but not function value
  • Match the type of discontinuity with its characteristic:
    Removable ↔️ Limit exists but doesn't match function value
    Jump ↔️ Different left and right limits
    Infinite ↔️ Function approaches infinity
  • Discontinuities are points where a function is not continuous.
  • Identifying and removing discontinuities is essential for analyzing the behavior of a function.
    True
  • Steps to remove a removable discontinuity by factoring:
    1️⃣ Original function
    2️⃣ Factor the numerator
    3️⃣ Cancel common factors
    4️⃣ Simplify the function
  • How many main types of discontinuities are there in a function?
    Three
  • Match the type of discontinuity with its example:
    Removable ↔️ f(x)=f(x) =x2−4x−2 \frac{x^{2} - 4}{x - 2}
    Jump ↔️ f(x)=f(x) = \begin{cases} x & \text{if } x < 0 \\ x + 1 & \text{if } x \geq 0 \end{cases}
    Infinite ↔️ f(x)=f(x) =1x \frac{1}{x}