LIMIT LAWS

• Limit laws


  Mathematical concepts that describe the behavior of functions as the input values approach a certain value
• Theorem 1.3 (i)


  lim x→c[k f(x)] = k lim x→c f(x)
• Theorem 1.3 (ii)


  lim x→c[f(x) ± g(x)] = lim x→c f(x) ± lim x→c g(x)
• Theorem 1.3 (iii)


  lim x→c[f(x)g(x)] = [lim x→c f(x)] [lim x→c g(x)]
• Theorem 1.3 (iv)


  lim x→c f(x)/g(x) = lim x→c f(x) / lim x→c g(x), if lim x→c g(x) ≠ 0
• (iv)


  1. Show that lim(x→c) 1/g(x) = 1/L2, for L2 ≠ 0

  2. Make |1/g(x) - 1/L2| as small as desired

  3. Numerator of fraction on right-hand side of (1.8) can be made as small as needed

  4. Consider what happens to the denominator

  5. For ε2 = |L2|/2, |g(x) - L2| < |L2|/2

  6. By triangle inequality, |L2| = |L2 - g(x) + g(x)| ≤ |L2 - g(x)| + |g(x)| < |L2|/2 + |g(x)|

  7. Subtracting |L2|/2 from both sides gives |L2|/2 < |g(x)|

  8. 1/|g(x)| < 2/|L2|

  9. From (1.8), for 0 < |x - c| < δ2, |1/g(x) - 1/L2| < 2|L2 - g(x)|/L2^2

  10. Given ε > 0, ∃ δ3 > 0 such that |L2 - g(x)| < εL2^2/2, whenever 0 < |x - c| < δ3

  11. For δ = min{δ2, δ3}, |1/g(x) - 1/L2| < ε, whenever 0 < |x - c| < δ
• Let n ∈ ℕ

  lim(x→c) [f(x)]^n = [lim(x→c) f(x)]^n
• Case 1: L > 0

  1. Show that lim(x→c) √f(x)^n = √L^n

  2. Given any ε > 0, ∃ δ > 0 such that -ε + √L^n < √f(x)^n < ε + √L^n, whenever 0 < |x - c| < δ

  3. Take ε1 = min{L - (-ε + √L^n)^n, (ε + √L^n)^n - L} > 0

  4. ∃ δ > 0 such that (−ε + √L^n)^n - L ≤ −ε1 < f(x) - L < ε1 ≤ (ε + √L^n)^n - L, whenever 0 < |x - c| < δ
• Case 2: L < 0 and n is odd

  1. Let g(x) = -f(x)

  2. lim(x→c) √g(x)^n = lim(x→c) -√f(x)^n = -√L^n

  3. lim(x→c) -√f(x)^n = -√lim(x→c) f(x)^n = -√L^n

  4. lim(x→c) √f(x)^n = √lim(x→c) f(x)^n = √L^n
• Case 3: L = 0 and n is odd

  1. ∀ ε1 > 0, ∃ δ > 0 such that |f(x)| < ε1, whenever 0 < |x - c| < δ

  2. |√f(x)^n - √L^n| = |√f(x)^n| = √|f(x)|^n < ε if and only if √|f(x)|^n < ε^n

  3. Take ε1 = ε^n implies √|f(x)|^n < ε1, whenever 0 < |x - c| < δ