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Limit laws with given limits practice problems

Combine known limits with the sum, product, quotient and power laws. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Suppose \( \displaystyle \lim_{x\to 3} f(x) = 9 \), \( \displaystyle \lim_{x\to 3} g(x) = -6 \) and \( \displaystyle \lim_{x\to 3} h(x) = 2 \). Find \( \displaystyle \lim_{x\to 3} \left(- f(x) + \left[h(x)\right]^{3}\right) \).

Problem 1.209easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{n\to\infty} a_n = 9 \), \( \displaystyle \lim_{n\to\infty} b_n = 4 \) and \( \displaystyle \lim_{n\to\infty} c_n = 4 \). Find \( \displaystyle \lim_{n\to\infty} \left(4 a_n - \frac{b_n}{c_n}\right) \).

Problem 1.210easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{x\to -4} f(x) = 3 \), \( \displaystyle \lim_{x\to -4} g(x) = 1 \) and \( \displaystyle \lim_{x\to -4} h(x) = -4 \). Find \( \displaystyle \lim_{x\to -4} \left(- f(x) + \left[h(x)\right]^{3}\right) \).

Problem 1.211easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{x\to 2} f(x) = 3 \), \( \displaystyle \lim_{x\to 2} g(x) = 5 \) and \( \displaystyle \lim_{x\to 2} h(x) = -3 \). Find \( \displaystyle \lim_{x\to 2} \left(\frac{f(x) g(x)}{h(x)}\right) \).

Problem 1.212easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{n\to\infty} a_n = -3 \), \( \displaystyle \lim_{n\to\infty} b_n = 2 \) and \( \displaystyle \lim_{n\to\infty} c_n = 6 \). Find \( \displaystyle \lim_{n\to\infty} \left(a_n b_n - c_n\right) \).

Problem 1.213easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{x\to 4} f(x) = 3 \), \( \displaystyle \lim_{x\to 4} g(x) = -5 \) and \( \displaystyle \lim_{x\to 4} h(x) = -4 \). Find \( \displaystyle \lim_{x\to 4} \left(2 f(x) + 3 g(x)\right) \).

Problem 1.214easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{x\to -4} f(x) = -1 \), \( \displaystyle \lim_{x\to -4} g(x) = -6 \) and \( \displaystyle \lim_{x\to -4} h(x) = 2 \). Find \( \displaystyle \lim_{x\to -4} \left(4 f(x) - \frac{g(x)}{h(x)}\right) \).

Problem 1.215easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{x\to 5} f(x) = 6 \), \( \displaystyle \lim_{x\to 5} g(x) = 1 \) and \( \displaystyle \lim_{x\to 5} h(x) = -2 \). Find \( \displaystyle \lim_{x\to 5} \left(\left(f(x) - g(x)\right)^{2}\right) \).

Problem 1.216easy✓ Nihil obstat

Suppose \( \displaystyle \lim_{x\to 4} f(x) = -6 \), \( \displaystyle \lim_{x\to 4} g(x) = 8 \) and \( \displaystyle \lim_{x\to 4} h(x) = 2 \). Find \( \displaystyle \lim_{x\to 4} \left(\left(f(x) - g(x)\right)^{2}\right) \).

Problem 1.217easy✓ Every equation proved

Suppose \( \displaystyle \lim_{x\to 0} f(x) = -5 \), \( \displaystyle \lim_{x\to 0} g(x) = 1 \) and \( \displaystyle \lim_{x\to 0} h(x) = 6 \). Find \( \displaystyle \lim_{x\to 0} \left(2 f(x) + 3 g(x)\right) \).

Problem 1.218easy✓ Nihil obstat