MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
Limits and Their Properties

1.5 Infinite Limits

Focus on compare infinite one-sided limits, signed infinite limits, vertical asymptotes versus holes, squared denominators.

Free preview Explore the lesson before creating an account.

Play games for this section

Before you solve

Ideas in this lesson

01Compare infinite one-sided limitsWorked steps · five questions02Signed infinite limitsWorked steps · five questions03Vertical asymptotes versus holesWorked steps · five questions04Squared denominatorsWorked steps · five questions
Idea 1 of 4

Compare infinite one-sided limits

Opposite infinite one-sided limits mean the two-sided limit does not exist. A squared denominator can send both sides to positive infinity.

Worked example 1

Find the two-sided limit as x→2x \to 2 of 1(x−2)\dfrac{1}{(x - 2)}.
  1. x−2→0−  ⟹  f(x)→−∞x - 2 \to 0^{-} \implies f(x) \to -\infty
  2. x−2→0+  ⟹  f(x)→∞x - 2 \to 0^{+} \implies f(x) \to \infty
  3. lim⁡\displaystyle \lim ₓ→2−f(x)≠lim⁡\displaystyle \to 2^{-} f(x) \ne \lim ₓ→2+f(x)  ⟹  DNE\to 2^{+} f(x) \implies DNE

Answer: Does not exist

Worked example 2

Find the two-sided limit as x→3x \to 3 of 1(x−3)2\dfrac{1}{(x - 3)^{2}}.
  1. x−3→0−  ⟹  f(x)→∞x - 3 \to 0^{-} \implies f(x) \to \infty
  2. x−3→0+  ⟹  f(x)→∞x - 3 \to 0^{+} \implies f(x) \to \infty
  3. lim⁡f(x)=∞\displaystyle \lim f(x) = \infty

Answer: Positive infinity

Idea 2 of 4

Signed infinite limits

Approaching a zero denominator requires a sign check. A reciprocal may grow toward positive or negative infinity.

Worked example 1

Evaluate lim⁡x→2+1x−2\displaystyle \lim\limits_{x\to 2^{+}} \dfrac{1}{x - 2}.
  1. x>2  ⟹  x−2>0x > 2 \implies x - 2 > 0
  2. x−2→0+x - 2 \to 0^{+}
  3. Reciprocal limit =∞= \infty

Answer: ∞\infty

Worked example 2

Evaluate lim⁡x→3−1x−3\displaystyle \lim\limits_{x\to 3^{-}} \dfrac{1}{x - 3}.
  1. x<3  ⟹  x−3<0x < 3 \implies x - 3 < 0
  2. x−3→0−x - 3 \to 0^{-}
  3. Reciprocal limit =−∞= -\infty

Answer: −∞-\infty

Idea 3 of 4

Vertical asymptotes versus holes

Cancel common factors only away from excluded inputs. Remaining denominator zeros may produce asymptotes; canceled zeros may produce holes.

Worked example 1

Find the vertical asymptote of f(x)=1x−2f(x) = \dfrac{1}{x - 2}.
  1. x−2=0  ⟹  x=2x - 2 = 0 \implies x = 2
  2. Numerator at x=2x = 2 is 1≠01 \ne 0
  3. Vertical asymptote: x=2x = 2

Answer: 22

Worked example 2

Find the vertical asymptote of f(x)=x+1x−3f(x) = \dfrac{x + 1}{x - 3}.
  1. x−3=0  ⟹  x=3x - 3 = 0 \implies x = 3
  2. Numerator at x=3x = 3 is 4≠04 \ne 0
  3. Vertical asymptote: x=3x = 3

Answer: 33

Idea 4 of 4

Squared denominators

A squared nonzero denominator is positive on either side. This can make both one-sided limits grow in the same direction.

Worked example 1

Evaluate lim⁡x→2+1x−2\displaystyle \lim\limits_{x\to 2^{+}} \dfrac{1}{x - 2}.
  1. x>2  ⟹  x−2>0x > 2 \implies x - 2 > 0
  2. x−2→0+x - 2 \to 0^{+}
  3. Reciprocal limit =∞= \infty

Answer: ∞\infty

Worked example 2

Evaluate lim⁡x→3−1x−3\displaystyle \lim\limits_{x\to 3^{-}} \dfrac{1}{x - 3}.
  1. x<3  ⟹  x−3<0x < 3 \implies x - 3 < 0
  2. x−3→0−x - 3 \to 0^{-}
  3. Reciprocal limit =−∞= -\infty

Answer: −∞-\infty

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §1.5,p1.5, p. 8383. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.