Mathematics
Grade 12
15 min
Find limits at vertical asymptotes using graphs
Find limits at vertical asymptotes using graphs
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1
Introduction & Learning Objectives
Learning Objectives
Identify the location of a vertical asymptote from a function's graph.
Determine the left-hand limit as x approaches a vertical asymptote by analyzing the graph's behavior.
Determine the right-hand limit as x approaches a vertical asymptote by analyzing the graph's behavior.
Use correct limit notation to describe a function's behavior as approaching positive or negative infinity.
Conclude whether the two-sided limit at a vertical asymptote is positive infinity, negative infinity, or does not exist (DNE).
Distinguish between a limit that is infinite and a limit that does not exist because the one-sided limits differ.
Sketch the behavior of a graph near a vertical asymptote given the one-sided limits.
Ever seen a graph that seems to sh...
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Key Concepts & Vocabulary
TermDefinitionExample
Vertical AsymptoteA vertical line, x = c, that the graph of a function approaches but never touches or crosses. It represents a value of x for which the function's output grows or shrinks without bound.For the function f(x) = 1/x, the y-axis (the line x = 0) is a vertical asymptote.
Infinite LimitA limit where the function's y-values increase or decrease without bound as x approaches a specific number. We say the limit is positive infinity (+∞) or negative infinity (-∞).As x gets closer to 0, the values of 1/x² get larger and larger. We write this as lim_{x→0} (1/x²) = +∞.
Left-Hand LimitThe value a function's output (y-value) approaches as the input (x-value) gets closer to a number 'c' from the left side (i.e., from values less than c).For...
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Core Formulas
One-Sided Infinite Limits from a Graph
1. To find lim_{x \to c^{-}} f(x), trace the graph from the left of x=c. If it goes up indefinitely, the limit is +∞. If it goes down, it is -∞.
2. To find lim_{x \to c^{+}} f(x), trace the graph from the right of x=c. If it goes up indefinitely, the limit is +∞. If it goes down, it is -∞.
Use this rule to determine the behavior of the function on each individual side of the vertical asymptote.
Two-Sided Infinite Limits from a Graph
If lim_{x \to c^{-}} f(x) = lim_{x \to c^{+}} f(x) = +∞, then lim_{x \to c} f(x) = +∞.
If lim_{x \to c^{-}} f(x) = lim_{x \to c^{+}} f(x) = -∞, then lim_{x \to c} f(x) = -∞.
This rule allows you to state the two-sided limit if and only if the function's behavior is the same (both rising or both fallin...
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Challenging
A function f(x) has vertical asymptotes at x = -3 and x = 3. The graph shows that lim_{x \to -3^{-}} f(x) = lim_{x \to 3^{+}} f(x) and lim_{x \to -3^{+}} f(x) = lim_{x \to 3^{-}} f(x) = -\infty. If lim_{x \to -3} f(x) does not exist, what is the value of lim_{x \to 3^{+}} f(x)?
A.-\infty
B.+\infty
C.0
D.Cannot be determined
Challenging
You are given the following information about a function g(x): it has a single vertical asymptote at x = 2; lim_{x \to 2^{-}} g(x) = -\infty; and lim_{x \to 2^{+}} g(x) = +\infty. Which of the following descriptions matches the appearance of the graph near x = 2?
A.The graph resembles a 'volcano', rising on both sides of x = 2.
B.The graph resembles a 'canyon', falling on both sides of x = 2.
C.To the left of x = 2, the graph goes down; to the right, it goes up.
D.To the left of x = 2, the graph goes up; to the right, it goes down.
Challenging
Consider a function where lim_{x \to c^{-}} f(x) = L and lim_{x \to c^{+}} f(x) = M. The statement 'the limit at x=c is infinite' is only a partially correct description if...
A.L = +\infty and M = -\infty
B.L and M are finite numbers.
C.L = M = +\infty
D.L = M = -\infty
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