Practice finding limits from expressions, tables, and graphs in words, and determine whether functions are continuous.
Read each problem carefully. Show your work and explain your reasoning when needed.
Evaluating limits and identifying continuity
Math - Grade 9-12
- 1
Evaluate the limit: lim as x approaches 3 of (2x + 5).
- 2
Evaluate the limit: lim as x approaches -2 of (x^2 - 4x + 1).
- 3
Evaluate the limit: lim as x approaches 4 of (x^2 - 16) / (x - 4).
- 4
Evaluate the limit: lim as x approaches 2 of (x^2 + x - 6) / (x - 2).
- 5
Evaluate the limit: lim as x approaches 0 of sin(x) / x.
- 6
Evaluate the limit: lim as x approaches infinity of 5 / x.
- 7
Evaluate the limit: lim as x approaches infinity of (3x^2 + 1) / (x^2 - 4).
- 8
Find the one-sided limits for f(x) = |x| / x at x = 0. State the left-hand limit and the right-hand limit, then say whether the two-sided limit exists.
- 9
A function is defined by f(x) = (x^2 - 1) / (x - 1) for x not equal to 1, and f(1) = 5. Is the function continuous at x = 1? Explain.
- 10
A function is defined by f(x) = x^2 for x less than or equal to 1, and f(x) = 2x - 1 for x greater than 1. Is f(x) continuous at x = 1? Explain.
- 11
Use the table values to estimate lim as x approaches 2 of f(x): when x = 1.9, f(x) = 3.8; when x = 1.99, f(x) = 3.98; when x = 2.01, f(x) = 4.02; when x = 2.1, f(x) = 4.2.
- 12
Determine whether the function f(x) = 1 / (x - 6) is continuous at x = 6. Explain.
- 13
Evaluate the limit: lim as x approaches -1 of (x^3 + 1) / (x + 1).
- 14
State whether each function is continuous everywhere on the real numbers: f(x) = 4x - 7, g(x) = x^2 + 3x + 1, and h(x) = 1 / (x + 2).
- 15
A graph has an open circle at (3, 7) and a filled point at (3, 4). The curve approaches y = 7 from both sides as x approaches 3. Find lim as x approaches 3 of f(x), and state f(3).