Limits worksheet with answers

Free printable Limits worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
xy9x = 9

Find the limit L = limx→9 (√x − 3)/(x − 9).

f(x) = (√x − 3)/(x − 9)x → 9
L =
2.Easy
xy1x = 1

Find the limit L = limx→1 (√x − 1)/(x − 1).

f(x) = (√x − 1)/(x − 1)x → 1
L =
3.Medium
xyy = L

Find the limit L = limx→∞ (−6x2 + x − 8)/(4x2 + 4x + 5).

f(x) = (−6x2 + x − 8)/(4x2 + 4x + 5)x → ∞
L =
4.Medium
xyy = L

Find the limit L = limx→∞ (1 + 3/x)2x. Give a decimal value.

f(x) = (1 + 3/x)2xx → ∞
L =
5.Hard
xyx = 0

Find the limit L = limx→0 (1 − cos 3x)/(2x²), with x in radians.

f(x) = (1 − cos 3x)/(2x²)x → 0
L =
6.Hard
xyx = 0

Find the limit L = limx→0 (1 − cos 2x)/(2x²), with x in radians.

f(x) = (1 − cos 2x)/(2x²)x → 0
L =

Answer key

  1. 1. L = 0.1667
  2. 2. L = 0.5
  3. 3. L = −1.5
  4. 4. L = 403.4
  5. 5. L = 2.25
  6. 6. L = 1

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