Exercises
Strengthen your understanding of limits in calculus with this quiz covering direct substitution, limit laws, limits at infinity, and indeterminate forms. Solve problems involving linear and polynomial functions, rational expressions, sums of functions, and the behavior of functions as x approaches positive or negative infinity. You will also practice essential special limits, including sin(x)/x, tan(x)/x, and (e^x - 1)/x as x approaches 0. Explore when algebraic simplification helps evaluate a limit and recognize cases where a two-sided limit does not exist. This quiz is ideal for students reviewing foundational differential calculus concepts.
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To find the limit of f(x) = 3x + 2 as x approaches 1, substitute x with 1: 3(1) + 2 = 5.
The limit of the sum is the sum of the limits: lim (x -> 2) [f(x) + g(x)] = lim (x -> 2) f(x) + lim (x -> 2) g(x) = 4 + 6 = 10.
As x approaches infinity, 1/x becomes smaller and tends towards 0.
This is a well-known limit in calculus. lim (x -> 0) (sin x) / x = 1.
Factor the numerator: (x - 1)(x + 1) / (x - 1). Cancel (x - 1) to get x + 1. Substituting x = 1 gives 2.
As x approaches -∞, x^2 grows larger towards Infinity.
Divide every term by x^2: (3 + 1/x). As x approaches infinity, 1/x approaches 0, so the limit is 3.
Factor the numerator: 2(x^2-3) = 2(x-3)(x+3)/(x-3). Cancel (x-3), to get 2(x+3). Substituting x = 3 gives 12.
This is a standard limit lim (x -> 0) (e^x - 1)/x = 1.
The limit, using the fact that tan(x) ≈ sin(x) + (x^3/3) near 0, is 1.
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