Practice evaluating limits algebraically using direct substitution, factoring, conjugates, standard trigonometric limits, and end behavior.

Read each problem carefully. Show the algebraic steps you use to evaluate each limit.

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Using substitution, factoring, rationalizing, and limit laws

Math - Grade 9-12

Instructions Read each problem carefully. Show the algebraic steps you use to evaluate each limit.
  1. 1

    Evaluate the limit as x approaches 3: 2x^2 - 5x + 1.

  2. 2

    Evaluate the limit as x approaches 3: (x^2 - 9)/(x - 3).

  3. 3

    Evaluate the limit as x approaches -2: (x^2 + 5x + 6)/(x + 2).

  4. 4

    Evaluate the limit as x approaches 4: (sqrt(x + 5) - 3)/(x - 4).

  5. 5

    Evaluate the limit as x approaches 5: (1/x - 1/5)/(x - 5).

  6. 6

    Evaluate the limit as x approaches 2: (x^3 - 8)/(x - 2).

  7. 7

    Evaluate the limit as x approaches 4: (x^2 - 4x)/(x^2 - 16).

  8. 8

    Evaluate the limit as h approaches 0: ((3 + h)^2 - 9)/h.

  9. 9

    Evaluate the limit as x approaches 0: sin(5x)/x.

  10. 10

    Evaluate the limit as x approaches 0: (1 - cos x)/x.

  11. 11

    Evaluate the limit as x approaches infinity: (5x^2 - 3x + 1)/(2x^2 + 7).

  12. 12

    Evaluate the limit as x approaches infinity: (4x^3 + x)/(2x^4 - 1).

  13. 13
    Unlabeled graph of a piecewise function with a curved left branch and a straight right branch meeting at one point.

    Let f(x) = x^2 + 2 for x < 1 and f(x) = 4x - 1 for x >= 1. Evaluate the limit of f(x) as x approaches 1.

  14. 14
    Unlabeled graph showing a jump discontinuity with different left and right horizontal values.

    Evaluate the limit as x approaches 2: |x - 2|/(x - 2).

  15. 15

    Evaluate the limit as x approaches 0: (sqrt(1 + x) - 1)/x.

LivePhysics™.com Math - Grade 9-12