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: limx→a−f(x)=limx→a+f(x)=limx→af(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.
  • Introduction
    No more finding limits “graphically”; Now, finding limits “algebraically”!
    a)
    What is Direct Substitution?

  • 1.
    Evaluate the limit:
    a)
    limx→3(5x2−20x+17)\lim_{x \to 3} (5x^2-20x+17)

    b)
    limx→−2x3+3x2−15−4x\lim_{x \to -2} \frac{{{x^3} + 3{x^2} - 1}}{{5 - 4x}}

    c)
    limx→0∣x∣\lim_{x \to 0}\left| x \right|

    d)
    limx→π2sinx2−cosx\lim_{x \to \frac{\pi }{2}} \;\frac{{\sin x}}{{2 - \cos x}}

  • 2.
    Evaluate the one-sided limit:
    a)
    limx→3−(5x2−20x+17)\lim_{x \to 3^-} (5x^2-20x+17)
    limx→3+(5x2−20x+17)\lim_{x \to 3^+} (5x^2-20x+17)

    b)
    limx→4−x−4\lim_{x \to {4^ - }} \sqrt {x - 4}
    limx→4+x−4\lim_{x \to {4^ + }} \sqrt {x - 4}

  • 3.

    Finding limits algebraically using direct substitution
  • 4.

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

    b)
    limx→4−g(x)\lim_{x \to {4^ - }} g(x)
    limx→4+g(x)\lim_{x \to {4^ + }} g(x)
    limx→4g(x)\lim_{x \to {4}} g(x)

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1.
Limits
1.1
Introduction to Calculus - Limits
1.2
Finding limits from graphs
1.3
Limit laws
1.4
Continuity
1.5
Finding limits algebraically - direct substitution
1.6
Finding limits algebraically - when direct substitution is not possible
1.7
Infinite limits - vertical asymptotes
1.8
Limits at infinity - horizontal asymptotes
1.9
Intermediate value theorem
1.10
Squeeze theorem
Calculus 1
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