Finding limits algebraically - when direct substitution is not possible

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  1. Simplify Out "Zero Denominator" by Cancelling Common Factors

    Find limx3  x29x3\lim_{x \to 3} \;\frac{{{x^2} - 9}}{{x - 3}}

    1. Expand First, Then Simplify Out "Zero Denominator" by Cancelling Common Factors

      Evaluate limh0  (5+h)225h\lim_{h \to 0} \;\frac{{{{\left( {5 + h} \right)}^2} - 25}}{h}

      1. Simplify Out "Zero Denominator" by Rationalizing Radicals


        1. limx4  4x2x\lim_{x \to 4} \;\frac{{4 - x}}{{2 - \sqrt x }}
          (hint: rationalize the denominator by multiplying its conjugate)
      2. Find Limits of Functions involving Absolute Value

        Evaluate limx0  xx\lim_{x \to 0} \;\frac{{\left| x \right|}}{x}

        (hint: express the absolute value function as a piece-wise function)

        1. Find Limits Using the Trigonometric Identity:limθ0  sin  θθ=1\lim_{\theta \to 0} \;\frac{{{sin\;}\theta}}{{\theta}}=1

          Find limx0  sin  5x2x\lim_{x \to 0} \;\frac{{{sin\;}5x}}{{2x}}

          Topic Notes
          There are times when applying direct substitution would only give us an undefined solution. In this section, we will explore some cool tricks to evaluate limits algebraically, such as using conjugates, trigonometry, common denominators, and factoring.