1.12 Confirming Continuity over an Interval

    Cards (98)

    • A function f(x)f(x) is continuous at x=x =a a if f(a)f(a) is defined
    • For a function to be continuous at x=x =a a, limxaf(x)\lim_{x \to a} f(x) must exist.
    • A function is continuous at x=x =a a if \lim_{x \to a} f(x) = f(a)</latex>, meaning the limit equals the function's value
    • Match the condition with its description:
      Function defined ↔️ f(a)f(a) must be a real number
      Limit exists ↔️ limxaf(x)\lim_{x \to a} f(x) must approach a single value from both sides
      Limit equals function value ↔️ limxaf(x)=\lim_{x \to a} f(x) =f(a) f(a)
    • The function f(x) = \frac{x^{2} - 4}{x - 2}</latex> is defined at x=x =2 2.

      False
    • The limit limx2x24x2\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} equals 4
    • The function f(x)=f(x) =x24x2 \frac{x^{2} - 4}{x - 2} is continuous at x=x =2 2.

      False
    • The function f(x)=f(x) =x24x2 \frac{x^{2} - 4}{x - 2} is undefined atx = 2</latex> because it results in 0/0
    • The function f(x)=f(x) =1x2 \frac{1}{x - 2} is undefined at x=x =2 2 because it results in division by zero
    • The limit limx21x2\lim_{x \to 2} \frac{1}{x - 2} exists.

      False
    • The function f(x)=f(x) =x23x+ x^{2} - 3x +2 2 is continuous at x=x =1 1 because \lim_{x \to 1} f(x) = f(1) = 0</latex>, satisfying all three conditions for continuity
    • The function f(x)=f(x) =x23x+ x^{2} - 3x +2 2 is continuous at x=x =1 1.
    • The function f(x)=f(x) = \begin{cases} x^{2} & \text{if } x \neq 1 \\ 2 & \text{if } x = 1 \end{cases} is discontinuous at x=x =1 1 because limx1f(x)f(1)\lim_{x \to 1} f(x) \neq f(1), failing the third condition for continuity
    • The piecewise function f(x) = \begin{cases} x^{2} & \text{if } x \neq 1 \\ 2 & \text{if } x = 1 \end{cases}</latex> is continuous at x=x =1 1.

      False
    • What is the condition for continuity that requires limxaf(x)=\lim_{x \to a} f(x) =f(a) f(a)?

      Limit equals function value
    • The limit of f(x)=f(x) =x2 x^{2} as x1x \to 1 is 1
    • The function f(x) = \begin{cases} x^{2} & \text{if } x \neq 1 \\ 2 & \text{if } x = 1 \end{cases}</latex> is continuous at x=x =1 1.

      False
    • Match the continuity condition with its description:
      Function defined ↔️ f(a)f(a) must be a real number
      Limit exists ↔️ limxaf(x)\lim_{x \to a} f(x) must exist
      Limit equals function value ↔️ limxaf(x)=\lim_{x \to a} f(x) =f(a) f(a)
    • For a function to be continuous at x=x =a a, the limit limxaf(x)\lim_{x \to a} f(x) must exist
    • The condition limxaf(x)=\lim_{x \to a} f(x) =f(a) f(a) ensures that the function value and the limit at x=x =a a are equal.
    • Conditions for a function to be continuous from the right at x=x =a a
      1️⃣ f(a)f(a) is defined
      2️⃣ limxa+f(x)\lim_{x \to a^ + } f(x) exists
      3️⃣ limxa+f(x)=\lim_{x \to a^ + } f(x) =f(a) f(a)
    • What type of continuity is checked at the right endpoint of an interval?
      Left continuity
    • A function is continuous from the right at x=x =a a if \lim_{x \to a^ + } f(x)</latex> equals f(a)
    • Match the type of continuity with its conditions:
      Right Continuity ↔️ 1. f(a)f(a) defined, 2. limxa+f(x)\lim_{x \to a^ + } f(x) exists, 3. limxa+f(x)=\lim_{x \to a^ + } f(x) =f(a) f(a)
      Left Continuity ↔️ 1. f(b)f(b) defined, 2. limxbf(x)\lim_{x \to b^ - } f(x) exists, 3. limxbf(x)=\lim_{x \to b^ - } f(x) =f(b) f(b)
    • Is the function f(x)=f(x) =x1 \sqrt{x - 1} continuous from the right at x=x =1 1?

      Yes
    • The function f(x)=f(x) =x1 \sqrt{x - 1} is continuous from the left atx = 5
    • A function continuous from the left at x = b</latex> must satisfy limxb+f(x)=\lim_{x \to b^ + } f(x) =f(b) f(b).

      False
    • The function f(x)=f(x) =x1 \sqrt{x - 1} is continuous from the right at x=x =1 1.
    • What is the value of f(5)f(5) for f(x)=f(x) =x1 \sqrt{x - 1}?

      2
    • A function is continuous over an open interval if it is continuous at every point within that interval
    • Is the polynomial f(x)=f(x) =x2+ x^{2} +2x3 2x - 3 continuous over ( - 1, 3)</latex>?

      Yes
    • A function is continuous at x=x =a a if f(a)f(a) is defined and limxaf(x)=\lim_{x \to a} f(x) =f(a) f(a).
    • What is the limit of x24x2\frac{x^{2} - 4}{x - 2} as x2x \to 2?

      4
    • The first condition for continuity at a point x=x =a a is that f(a)f(a) must be defined
    • The limit limxaf(x)\lim_{x \to a} f(x) must exist for continuity at x=x =a a.
    • What is the third condition for continuity at a point x=x =a a?

      limxaf(x)=\lim_{x \to a} f(x) =f(a) f(a)
    • The function value f(a)f(a) must be a real number
    • For the limit limxaf(x)\lim_{x \to a} f(x) to exist, it must approach a single value from both sides.
    • What must the limit limxaf(x)\lim_{x \to a} f(x) equal for continuity at x=x =a a?

      f(a)f(a)
    • In the example f(x) = \frac{x^{2} - 4}{x - 2}</latex>, f(2)f(2) is undefined
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