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)

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

b)

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

c)

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

d)

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

2.

Evaluate Infinite Limits Algebraically Find:

a)

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

b)

3.

4.

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

Infinite limits - vertical asymptotes

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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:

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

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