Derivatives - Other Indeterminant forms of L’Hopital’s Rule

• Recap of L’Hopital’s Rule
• Traditional form: 0/0
• Other indeterminant forms:
  • ∞/∞
  • 0 * ∞
  • ∞ - ∞
  • 0^0
  • ∞^0

Other Indeterminant Forms: ∞/∞

• Example 1:

  • lim (x → ∞) (x^2 + 3x)/(5x - 2)
  • Apply L’Hopital’s rule: lim (x → ∞) (2x + 3)/(5)
  • Final answer: ∞/5

• Example 2:

  • lim (x → ∞) (2x^3 - 5x^2)/(3x^3 - 8x - 2)
  • Apply L’Hopital’s rule: lim (x → ∞) (6x^2 - 10x)/(9x^2 - 8)
  • Final answer: 6/9 = 2/3

Other Indeterminant Forms: 0 * ∞

• Example 1:

  • lim (x → ∞) (x * ln(x))
  • Rewrite as: lim (x → ∞) (ln(x))/(1/x)
  • Apply L’Hopital’s rule: lim (x → ∞) (1/x)/(−1/x^2)
  • Final answer: lim (x → ∞) (x) = ∞

• Example 2:

  • lim (x → 0) (x * sin(1/x))
  • Rewrite as: lim (x → 0) (sin(1/x))/(1/x)
  • Apply L’Hopital’s rule: lim (x → 0) (cos(1/x) * (−1/x^2))/(−1/x^2)
  • Final answer: lim (x → 0) cos(1/x) does not exist

Other Indeterminant Forms: ∞ - ∞

• Example 1:

  • lim (x → ∞) (x^2 - x)
  • Rewrite as: lim (x → ∞) (1/x^2 - 1/x)
  • Apply L’Hopital’s rule: lim (x → ∞) (−2/x^3 + 1/x^2)
  • Final answer: lim (x → ∞) (−2/x^3 + 1/x^2) = 0

• Example 2:

  • lim (x → ∞) (x^3 - √x)
  • Rewrite as: lim (x → ∞) (1/x - 1/√x)
  • Apply L’Hopital’s rule: lim (x → ∞) (−1/x^2 - −1/2x^3)
  • Final answer: lim (x → ∞) (−1/x^2 + 1/2x^3) = 0

Other Indeterminant Forms: 0^0

• Example 1:

  • lim (x → 0) (x^x)
  • Rewrite as: lim (x → 0) (e^(xlnx))
  • Apply L’Hopital’s rule: lim (x → 0) (e^(lnx + x/x))
  • Final answer: lim (x → 0) (e^(lnx)) = 1

• Example 2:

  • lim (x → 0) (x^sinx)
  • Rewrite as: lim (x → 0) (e^(sinxlnx))
  • Apply L’Hopital’s rule: lim (x → 0) (e^((cosxlnx + sinx/x)))
  • Final answer: lim (x → 0) (e^(cosxlnx)) = 1

Other Indeterminant Forms: ∞^0

• Example 1:

  • lim (x → ∞) (x^0)
  • Rewrite as: lim (x → ∞) (e^(0lnx))
  • Apply L’Hopital’s rule: lim (x → ∞) (e^(1/xlnx))
  • Final answer: lim (x → ∞) (e^(1/xlnx)) = 1

• Example 2:

  • lim (x → ∞) (x^(1/lnx))
  • Rewrite as: lim (x → ∞) (e^(lnx^(1/lnx)))
  • Apply L’Hopital’s rule: lim (x → ∞) (e^((1/lnx) * (1/lnx) / x))
  • Final answer: lim (x → ∞) (e^(1/∞)) = 1

Recap of L’Hopital’s Rule

• L’Hopital’s rule allows us to evaluate limits involving indeterminant forms.
• Indeterminant forms: 0/0, ∞/∞, 0 * ∞, ∞ - ∞, 0^0, ∞^0.
• Apply L’Hopital’s rule by taking the derivative of the numerator and denominator.
• Repeat the process until an indeterminant form is resolved or an answer is obtained.
• L’Hopital’s rule is applicable when the limit exists, and both numerator and denominator approach zero or infinity.

Traditional Indeterminant Form: 0/0

• Example 1:

  • lim (x → 1) (x^2 - 1)/(x - 1)
  • Apply L’Hopital’s rule: lim (x → 1) (2x)/(1)
  • Final answer: 2

• Example 2:

  • lim (x → 0) (sinx)/x
  • Apply L’Hopital’s rule: lim (x → 0) (cosx)/(1)
  • Final answer: 1

Traditional Indeterminant Form: 0/0 (Continued)

• Example 3:

  • lim (x → π/2) (sinx)/(cosx - 1)
  • Apply L’Hopital’s rule: lim (x → π/2) (cosx)/(-sinx)
  • Final answer: lim (x → π/2) (cosx)/(-sinx) does not exist

• Example 4:

  • lim (x → 0) (e^x - 1)/x
  • Apply L’Hopital’s rule: lim (x → 0) (e^x)/(1)
  • Final answer: 1

Traditional Indeterminant Form: 0/0 (Continued)

• Example 5:
  • lim (x → 0) (1 - cosx)/x^2
  • Rewrite as: lim (x → 0) (2sin^2(x/2))/(x^2)
  • Apply L’Hopital’s rule: lim (x → 0) (2sin(x))/(2x)
  • Final answer: 1

Other Indeterminant Forms: 0 * ∞

• Example 1:
  • lim (x → ∞) (e^x * ln(x))
  • Apply L’Hopital’s rule: lim (x → ∞) (e^x)/(1/x)
  • Final answer: lim (x → ∞) (e^x) = ∞
• Example 2:
  • lim (x → ∞) (x * ln(x^2))
  • Rewrite as: lim (x → ∞) (ln(x))/(1/x^2)
  • Apply L’Hopital’s rule: lim (x → ∞) (1/x)/(−2/x^3)
  • Final answer: lim (x → ∞) (-2/x^3) = 0

Other Indeterminant Forms: ∞ - ∞

• Example 1:
  • lim (x → ∞) (x^2 - x)
  • Rewrite as: lim (x → ∞) (1/x^2 - 1/x)
  • Apply L’Hopital’s rule: lim (x → ∞) (−2/x^3 + 1/x^2)
  • Final answer: lim (x → ∞) (−2/x^3 + 1/x^2) = 0
• Example 2:
  • lim (x → ∞) (x^3 - √x)
  • Rewrite as: lim (x → ∞) (1/x - 1/√x)
  • Apply L’Hopital’s rule: lim (x → ∞) (−1/x^2 - −1/2x^3)
  • Final answer: lim (x → ∞) (−1/x^2 + 1/2x^3) = 0

Other Indeterminant Forms: 0^0

• Example 1:
  • lim (x → 0) (x^x)
  • Rewrite as: lim (x → 0) (e^(xlnx))
  • Apply L’Hopital’s rule: lim (x → 0) (e^(lnx + x/x))
  • Final answer: lim (x → 0) (e^(lnx)) = 1
• Example 2:
  • lim (x → 0) (x^sinx)
  • Rewrite as: lim (x → 0) (e^(sinxlnx))
  • Apply L’Hopital’s rule: lim (x → 0) (e^((cosxlnx + sinx/x)))
  • Final answer: lim (x → 0) (e^(cosxlnx)) = 1

Other Indeterminant Forms: ∞^0

• Example 1:
  • lim (x → ∞) (x^0)
  • Rewrite as: lim (x → ∞) (e^(0lnx))
  • Apply L’Hopital’s rule: lim (x → ∞) (e^(1/xlnx))
  • Final answer: lim (x → ∞) (e^(1/xlnx)) = 1
• Example 2:
  • lim (x → ∞) (x^(1/lnx))
  • Rewrite as: lim (x → ∞) (e^(lnx^(1/lnx)))
  • Apply L’Hopital’s rule: lim (x → ∞) (e^((1/lnx) * (1/lnx) / x))
  • Final answer: lim (x → ∞) (e^(1/∞)) = 1

Recap of L’Hopital’s Rule

• L’Hopital’s rule allows us to evaluate limits involving indeterminant forms.
• Indeterminant forms: 0/0, ∞/∞, 0 * ∞, ∞ - ∞, 0^0, ∞^0.
• Apply L’Hopital’s rule by taking the derivative of the numerator and denominator.
• Repeat the process until an indeterminant form is resolved or an answer is obtained.
• L’Hopital’s rule is applicable when the limit exists, and both numerator and denominator approach zero or infinity.

Traditional Indeterminant Form: 0/0

• Example 1:
  • lim (x → 1) (x^2 - 1)/(x - 1)
  • Apply L’Hopital’s rule: lim (x → 1) (2x)/(1)
  • Final answer: 2
• Example 2:
  • lim (x → 0) (sinx)/x
  • Apply L’Hopital’s rule: lim (x → 0) (cosx)/(1)
  • Final answer: 1

Traditional Indeterminant Form: 0/0 (Continued)

• Example 3:
  • lim (x → π/2) (sinx)/(cosx - 1)
  • Apply L’Hopital’s rule: lim (x → π/2) (cosx)/(-sinx)
  • Final answer: lim (x → π/2) (cosx)/(-sinx) does not exist
• Example 4:
  • lim (x → 0) (e^x - 1)/x
  • Apply L’Hopital’s rule: lim (x → 0) (e^x)/(1)
  • Final answer: 1

Traditional Indeterminant Form: 0/0 (Continued)

• Example 5:
  • lim (x → 0) (1 - cosx)/x^2
  • Rewrite as: lim (x → 0) (2sin^2(x/2))/(x^2)
  • Apply L’Hopital’s rule: lim (x → 0) (2sin(x))/(2x)
  • Final answer: 1

Traditional Indeterminant Form: 0/0 (Continued)

• Example 6:
  • lim (x → 0) ((e^x - 1) / x) * (1 - cos(x))
  • Apply L’Hopital’s rule: lim (x → 0) (e^x)/(1) * sin(x)
  • Final answer: 0

Summary

• L’Hopital’s rule is a powerful tool for evaluating limits.
• It can be applied to other indeterminant forms such as ∞/∞, 0 * ∞, ∞ - ∞, 0^0, ∞^0.
• The key is to recognize the form and appropriately differentiate the numerator and denominator.
• With practice, using L’Hopital’s rule becomes easier and helps solve a wide range of limit problems.

21. Other Indeterminant Forms: ∞/∞

• Example 1:
  • lim (x → ∞) (x^3 + 2x^2)/(x^2 + 3x)
  • Apply L’Hopital’s rule: lim (x → ∞) (3x^2 + 4x)/(2x + 3)
  • Final answer: ∞/2 = ∞
• Example 2:
  • lim (x → ∞) (2x^2 + 3√x)/(4x^2 - 5x)
  • Apply L’Hopital’s rule: lim (x → ∞) (4x + (3/(2√x)))/(8x - 5)
  • Final answer: lim (x → ∞) (4x)/(8x) = 1/2

22. Other Indeterminant Forms: 0 * ∞

• Example 1:
  • lim (x → ∞) (x^2 * e^(1/x))
  • Apply L’Hopital’s rule: lim (x → ∞) (2x * e^(1/x) - x^2 * e^(1/x) / x^2)
  • Final answer: lim (x → ∞) (2 * e^(1/x) - 1) = 1
• Example 2:
  • lim (x → 0) (x * ln(x^3))
  • Rewrite as: lim (x → 0) (ln(x))/((1/x^3))
  • Apply L’Hopital’s rule: lim (x → 0) (1/x)/(−3/x^4)
  • Final answer: lim (x → 0) (−x/3) = 0

23. Other Indeterminant Forms: ∞ - ∞

• Example 1:
  • lim (x → ∞) (x^3 - x^2 - 2x)/(2x - 3)
  • Apply L’Hopital’s rule: lim (x → ∞) (3x^2 - 2x - 2)/(2)
  • Final answer: (3∞^2 - 2∞ - 2)/2 = ∞
• Example 2:
  • lim (x → ∞) (x^4 - √x - 1)/(2x - 5)
  • Rewrite as: lim (x → ∞) (1 - 1/√x - 1/x^4)/(2 - 5/x)
  • Apply L’Hopital’s rule: lim (x → ∞) (1/(2x^2) + 4/x^5)/(5/x^2)
  • Final answer: (1/(2∞^2) + 4/∞^5)/(5/∞^2) = 0

24. Other Indeterminant Forms: 0^0

• Example 1:
  • lim (x → 0) (x^x * (1 + x))
  • Apply L’Hopital’s rule: lim (x → 0) ((1 + ln(x)) * x^x + x^x)/(1 + x)
  • Final answer: lim (x → 0) ((1 + ln(x)) * x^x + x^x)/(1 + x) = 1
• Example 2:
  • lim (x → 0) ((1/x)^(1/x) * (ln(1/x) + 1))
  • Rewrite as: lim (x → 0) (e^(-ln(x)/x) * (-ln(x)/x^2 - 1/x^2 + 1))