Practice evaluating limits algebraically and graphically, identifying discontinuities, and applying the formal ideas of continuity.
Read each problem carefully. Show your work, justify limit steps, and state when a limit or function value does not exist.
Evaluating limits, one-sided behavior, and continuity
Math - Grade advanced
- 1
Evaluate the limit: lim as x approaches 3 of (x^2 - 9)/(x - 3).
- 2
Evaluate the limit: lim as x approaches 0 of sin(5x)/x.
- 3
Evaluate the limit: lim as x approaches 2 of (x^3 - 8)/(x - 2).
- 4
Evaluate the one-sided limits for f(x) = |x|/x as x approaches 0 from the left and from the right. Then state whether lim as x approaches 0 of f(x) exists.
- 5
Determine whether the function f(x) = (x^2 - 4)/(x - 2) is continuous at x = 2 if f(2) is defined to be 5.
- 6
Find the value of k that makes the piecewise function continuous at x = 1: f(x) = x^2 + k for x less than 1, and f(x) = 3x + 1 for x greater than or equal to 1.
- 7
Evaluate the limit: lim as x approaches infinity of (4x^2 - 3x + 1)/(2x^2 + 5).
- 8
Evaluate the limit: lim as x approaches infinity of (7x - 2)/(x^2 + 1).
- 9
A graph of f has an open circle at (2, 4), a filled point at (2, 1), and the curve approaches y = 4 from both sides of x = 2. Find lim as x approaches 2 of f(x), f(2), and state whether f is continuous at x = 2.
- 10
Use direct substitution to evaluate lim as x approaches -1 of (2x^3 - x + 4).
- 11
Evaluate the limit: lim as x approaches 0 of (1 - cos x)/x.
- 12
Classify the discontinuity at x = 3 for f(x) = (x + 1)/(x - 3).
- 13
Evaluate lim as x approaches 4 of (sqrt(x) - 2)/(x - 4).
- 14
Determine whether the Intermediate Value Theorem guarantees a solution to x^3 + x - 1 = 0 on the interval [0, 1].
- 15
For the piecewise function f(x) = 2x + 1 for x less than 0, f(x) = k for x = 0, and f(x) = x^2 + 1 for x greater than 0, find k so that f is continuous at x = 0.