1.5 Determining Limits Using Algebraic Properties

    Cards (106)

    • What do the Properties of Limits allow us to do?
      Simplify complex limits
    • The limit of a sum or difference is the sum or difference of the individual limits
    • Constants can be moved outside the limit in the constant multiple law.
    • What is the Product Law for limits?
      limxc[f(x)g(x)]=\lim_{x \to c} [f(x) \cdot g(x)] =limxcf(x)limxcg(x) \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)
    • The limit of a quotient is the quotient of the limits, provided the denominator's limit is not zero
    • The limit of a function raised to a power is the limit raised to that power.
    • What is the limit of a constant as xx approaches cc?

      limxck=\lim_{x \to c} k =k k
    • Match the limit law with its description:
      Sum and Difference ↔️ Splits limits of sums or differences
      Product Law ↔️ Splits limits of products
    • What is the value of \lim_{x \to 2} (x^{2} + 3x)</latex>?
      10
    • The limit of a product can be split into the product of individual limits.
    • The value of \lim_{x \to 3} (x \cdot \sqrt{x + 1})</latex> is 6
    • Steps to apply the sum and difference laws
      1️⃣ Split the limit into individual limits
      2️⃣ Evaluate each limit separately
      3️⃣ Add or subtract the results
    • What is the Quotient Law for limits?
      \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}</latex>
    • The value of \lim_{x \to 1} \frac{x^{2} + 1}{2x - 1} is 2
    • The product law states that constants can be moved outside the limit.
      False
    • What does the Constant Multiple Law for limits state?
      \lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)</latex>
    • The value of limx35x2\lim_{x \to 3} 5x^{2} is 45
    • The limit of a product is the product of the individual limits.
    • What is the value of \lim_{x \to 1} (x \cdot \sqrt{x + 3})</latex>?
      2
    • The value of \lim_{x \to 2} \frac{x^{2} + 1}{3x - 2} is \frac{5}{4}
    • What does the Power Law for limits state?
      \lim_{x \to c} [f(x)]^{n} = \left[ \lim_{x \to c} f(x) \right]^{n}</latex>
    • The limit of a constant is the constant itself
    • What is the limit of a constant value kk as xx approaches cc?

      limxck=\lim_{x \to c} k =k k
    • The limit of a sum or difference is the sum or difference of the individual limits
    • What does the constant multiple property of limits state?
      limxc[kf(x)]=\lim_{x \to c} [k \cdot f(x)] =klimxcf(x) k \cdot \lim_{x \to c} f(x)
    • The limit of a product is the product of the limits.
    • The limit of a quotient is the quotient of the limits, provided the denominator's limit is not zero
    • What is the power property of limits?
      limxc[f(x)]n=\lim_{x \to c} [f(x)]^{n} =[limxcf(x)]n \left[ \lim_{x \to c} f(x) \right]^{n}
    • Steps to apply the limit laws for sums, differences, products, and quotients
      1️⃣ Separate the limit using the sum or difference law
      2️⃣ Apply the constant multiple law if necessary
      3️⃣ Split products into individual limits using the product law
      4️⃣ Divide quotients into individual limits using the quotient law
    • The quotient law cannot be applied if the limit of the denominator is zero.
    • Direct substitution can be used to find limits when the function is continuous
    • For what type of functions is direct substitution applicable?
      Polynomial and rational functions
    • If direct substitution results in \frac{0}{0}</latex>, other techniques are required.
    • Steps to evaluate limits by factoring and canceling
      1️⃣ Factor the numerator and denominator
      2️⃣ Cancel common factors
      3️⃣ Substitute the value the variable approaches
    • When can you use direct substitution to evaluate a limit?
      Function is continuous
    • Direct substitution works for simple rational functions when the denominator is zero at the point of interest.
      False
    • Evaluate \lim_{x \to 3} (x^{2} - 2x + 5)</latex> using direct substitution.8
    • Evaluate limx2x+1x3\lim_{x \to 2} \frac{x + 1}{x - 3} using direct substitution.

      -3
    • Indeterminate forms such as 00\frac{0}{0} require factoring or rationalizing to evaluate the limit.
    • Steps for evaluating limits by factoring and canceling terms.
      1️⃣ Factor the numerator and denominator
      2️⃣ Cancel common factors
      3️⃣ Substitute the variable's approach value