Infinite limits - vertical asymptotes - Limits

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

Notes:

i)
${\;}\lim_{x \to {a^ - }} f\left( x \right) =\infty$ ii)
$\lim_{x \to {a^ + }} f\left( x \right) =\infty$ iii)
$\lim_{x \to {a^ - }} f\left( x \right) =,- \infty$ iv)
$\lim_{x \to {a^ + }} f\left( x \right) =,- \infty$

Intro Lesson

Introduction to Vertical Asymptotes
1.
Determine Infinite Limits Graphically

For the function $f$ whose graph is shown, state the following:
2.
Evaluate Infinite Limits Algebraically
Find:

Infinite limits - vertical asymptotes

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Lessons

Notes:

Intro Lesson

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 For the function fff whose graph is shown, state the following:

a)

lim⁡x→−4−f(x)\lim_{x \to - {4^ - }} \;f\left( x \right)limx→−4−​f(x) lim⁡x→−4+f(x)\lim_{x \to - {4^ + }} \;f\left( x \right)limx→−4+​f(x) lim⁡x→−4f(x)\lim_{x \to - 4} \;f\left( x \right)limx→−4​f(x)

b)

lim⁡x→1−f(x)\lim_{x \to {1^ - }} \;f\left( x \right)limx→1−​f(x) lim⁡x→1+f(x)\lim_{x \to {1^ + }} \;f\left( x \right)limx→1+​f(x) lim⁡x→1f(x)\lim_{x \to 1} \;f\left( x \right)limx→1​f(x)

c)

lim⁡x→3−f(x)\lim_{x \to {3^ - }} \;f\left( x \right)limx→3−​f(x) lim⁡x→3+f(x)\lim_{x \to {3^ + }} \;f\left( x \right)limx→3+​f(x) lim⁡x→3f(x)\lim_{x \to 3} \;f\left( x \right)limx→3​f(x)

d)

lim⁡x→5−f(x)\lim_{x \to {5^ - }} \;f\left( x \right)limx→5−​f(x) lim⁡x→5+f(x)\lim_{x \to {5^ + }} \;f\left( x \right)limx→5+​f(x) lim⁡x→5f(x)\lim_{x \to 5} \;f\left( x \right)limx→5​f(x)

2.

Evaluate Infinite Limits Algebraically Find:

a)

lim⁡x→0−1x\lim_{x \to {0^ - }} \;\frac{1}{x}limx→0−​x1​ lim⁡x→0+1x\lim_{x \to {0^ + }} \;\frac{1}{x}limx→0+​x1​ lim⁡x→01x\lim_{x \to 0} \;\frac{1}{x}limx→0​x1​

b)

lim⁡x→0−1x2\lim_{x \to {0^ - }} \;\frac{1}{{{x^2}}}limx→0−​x21​ lim⁡x→0+1x2\lim_{x \to {0^ + }} \;\frac{1}{{{x^2}}}limx→0+​x21​ lim⁡x→01x2\lim_{x \to 0} \;\frac{1}{{{x^2}}}limx→0​x21​

3.

Evaluate Limits Algebraically Find: lim⁡x→2−5xx−2\lim_{x \to {2^ - }} \;\frac{{5x}}{{x - 2}}limx→2−​x−25x​ lim⁡x→2+5xx−2\lim_{x \to {2^ + }} \;\frac{{5x}}{{x - 2}}limx→2+​x−25x​ lim⁡x→25xx−2\lim_{x \to 2} \;\frac{{5x}}{{x - 2}}limx→2​x−25x​

4.

Determine Infinite Limits of Log Functions Determine: lim⁡x→0+ln⁡x\lim_{x \to {0^ + }} \ln xlimx→0+​lnx

Infinite limits - vertical asymptotes

Don't just watch, practice makes perfect.

We have over 350 practice questions in Calculus for you to master.

or

Determine Infinite Limits Graphically

For the function $f$ whose graph is shown, state the following:

$\lim_{x \to - {4^ - }} \;f\left( x \right)$
$\lim_{x \to - {4^ + }} \;f\left( x \right)$
$\lim_{x \to - 4} \;f\left( x \right)$

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

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

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

Evaluate Infinite Limits Algebraically
Find:

$\lim_{x \to {0^ - }} \;\frac{1}{x}$
$\lim_{x \to {0^ + }} \;\frac{1}{x}$
$\lim_{x \to 0} \;\frac{1}{x}$

$\lim_{x \to {0^ - }} \;\frac{1}{{{x^2}}}$
$\lim_{x \to {0^ + }} \;\frac{1}{{{x^2}}}$
$\lim_{x \to 0} \;\frac{1}{{{x^2}}}$

Evaluate Limits Algebraically
Find:
$\lim_{x \to {2^ - }} \;\frac{{5x}}{{x - 2}}$
$\lim_{x \to {2^ + }} \;\frac{{5x}}{{x - 2}}$
$\lim_{x \to 2} \;\frac{{5x}}{{x - 2}}$

Determine Infinite Limits of Log Functions
Determine:
$\lim_{x \to {0^ + }} \ln x$