Infinite limits - vertical asymptotes

Infinite limits - vertical asymptotes

Limits don't always necessarily give numerical solutions. What happens if we take the limit of a function near its vertical asymptotes? We will answer this question in this section, as well as exploring the idea of infinite limits using one-sided limits and two-sided limits.

Lessons

i)
limxaf(x)={\;}\lim_{x \to {a^ - }} f\left( x \right) =\infty
ii)
limxa+f(x)=\lim_{x \to {a^ + }} f\left( x \right) =\infty
iii)
limxaf(x)=,\lim_{x \to {a^ - }} f\left( x \right) =,- \infty
iv)
limxa+f(x)=,\lim_{x \to {a^ + }} f\left( x \right) =,- \infty
Infinite limits - vertical asymptotes, x approaching a^- Infinite limits - vertical asymptotes, x approaching a^+ Infinite limits - vertical asymptotes, x approaching a^- Infinite limits - vertical asymptotes, x approaching a^+
  • Introduction
    Introduction to Vertical Asymptotes
    a)
    finite limits VS. infinite limits

    b)
    infinite limits translate to vertical asymptotes on the graph of a function

    c)
    vertical asymptotes and curve sketching


  • 1.
    Determine Infinite Limits Graphically
    Finding limits algebraically using direct substitution
    For the function ff whose graph is shown, state the following:
    a)
    limx4f(x)\lim_{x \to - {4^ - }} \;f\left( x \right)
    limx4+f(x)\lim_{x \to - {4^ + }} \;f\left( x \right)
    limx4f(x)\lim_{x \to - 4} \;f\left( x \right)

    b)
    limx1f(x)\lim_{x \to {1^ - }} \;f\left( x \right)
    limx1+f(x)\lim_{x \to {1^ + }} \;f\left( x \right)
    limx1f(x)\lim_{x \to 1} \;f\left( x \right)

    c)
    limx3f(x)\lim_{x \to {3^ - }} \;f\left( x \right)
    limx3+f(x)\lim_{x \to {3^ + }} \;f\left( x \right)
    limx3f(x)\lim_{x \to 3} \;f\left( x \right)

    d)
    limx5f(x)\lim_{x \to {5^ - }} \;f\left( x \right)
    limx5+f(x)\lim_{x \to {5^ + }} \;f\left( x \right)
    limx5f(x)\lim_{x \to 5} \;f\left( x \right)


  • 2.
    Evaluate Infinite Limits Algebraically
    Find:
    a)
    limx01x\lim_{x \to {0^ - }} \;\frac{1}{x}
    limx0+1x\lim_{x \to {0^ + }} \;\frac{1}{x}
    limx01x\lim_{x \to 0} \;\frac{1}{x}

    b)
    limx01x2\lim_{x \to {0^ - }} \;\frac{1}{{{x^2}}}
    limx0+1x2\lim_{x \to {0^ + }} \;\frac{1}{{{x^2}}}
    limx01x2\lim_{x \to 0} \;\frac{1}{{{x^2}}}


  • 3.
    Evaluate Limits Algebraically
    Find:
    limx25xx2\lim_{x \to {2^ - }} \;\frac{{5x}}{{x - 2}}
    limx2+5xx2\lim_{x \to {2^ + }} \;\frac{{5x}}{{x - 2}}
    limx25xx2\lim_{x \to 2} \;\frac{{5x}}{{x - 2}}

  • 4.
    Determine Infinite Limits of Log Functions
    Determine:
    limx0+lnx\lim_{x \to {0^ + }} \ln x