Finding limits algebraically - direct substitution

Finding limits algebraically - direct substitution

Graphically finding the limit of a function is not always easy, as an alternative, we now shift our focus to finding the limit of a function algebraically. In this section, we will learn how to apply direct substitution to evaluate the limit of a function.

Lessons

• if: a function ff is continuous at a number aa
then: direct substitution can be applied: limxaf(x)=limxa+f(x)=limxaf(x)=f(a)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) =\lim_{x \to a} f(x)= f(a)

• Polynomial functions are continuous everywhere, therefore “direct substitution” can ALWAYS be applied to evaluate limits at any number.
  • 1.
    No more finding limits “graphically”; Now, finding limits “algebraically”!
    a)
    What is Direct Substitution?

    b)
    When to apply Direct Substitution, and why Direct Substitution makes sense.
    Exercise: f(x)=1x2f(x)=\frac{1}{x-2}
    i) Find the following limits from the graph of the function.
    limx3f(x)\lim_{x \to 3} f(x)
    limx2.5f(x)\lim_{x \to 2.5} f(x)
    limx0f(x)\lim_{x \to 0} f(x)
    ii) Evaluate:
    f(3)f(3)
    f(2.5)f(2.5)
    f(0)f(0)


  • 2.
    Evaluate the limit:
    a)
    limx3(5x220x+17)\lim_{x \to 3} (5x^2-20x+17)

    b)
    limx2x3+3x2154x\lim_{x \to -2} \frac{{{x^3} + 3{x^2} - 1}}{{5 - 4x}}

    c)
    limx0x\lim_{x \to 0}\left| x \right|

    d)
    limxπ2sinx2cosx\lim_{x \to \frac{\pi }{2}} \;\frac{{\sin x}}{{2 - \cos x}}


  • 3.
    Evaluate the one-sided limit:
    a)
    limx3(5x220x+17)\lim_{x \to 3^-} (5x^2-20x+17)
    limx3+(5x220x+17)\lim_{x \to 3^+} (5x^2-20x+17)

    b)
    limx4x4\lim_{x \to {4^ - }} \sqrt {x - 4}
    limx4+x4\lim_{x \to {4^ + }} \sqrt {x - 4}


  • 4.

    Finding limits algebraically using direct substitution

  • 5.

    Finding limits of a function algebraically by direct substitution
    a)
    limx1g(x)\lim_{x \to {-1^ - }} g(x)
    limx1+g(x)\lim_{x \to {-1^ + }} g(x)
    limx1g(x)\lim_{x \to {-1}} g(x)

    b)
    limx4g(x)\lim_{x \to {4^ - }} g(x)
    limx4+g(x)\lim_{x \to {4^ + }} g(x)
    limx4g(x)\lim_{x \to {4}} g(x)