Calculus Volume 1 · free preview
The Limit Laws
Limits combine according to algebraic laws, sums, products, quotients, powers and roots pass through limits, and together with substitution these laws evaluate most limits without tables or graphs.
1. The basic laws
The basic laws
2. Substitution and its limits
Substitution and its limits
3. Algebraic moves for 0/0
Algebraic moves for 0/0
4. Limits at infinity
Limits at infinity
5. Sum law in action
Given lim x to 2 of f equals 3 and lim x to 2 of g equals 5, what is lim x to 2 of (f + g)?
6. Quotient law condition
The quotient law says lim of f/g equals lim f over lim g under one essential condition. What is it?
7. Factoring a rational limit
What is lim x to 3 of (x^2 - 9)/(x - 3)?
8. Conjugate move
To evaluate lim x to 0 of (sqrt(x+1) - 1)/x, which algebraic step resolves the indeterminate form?
9. Horizontal asymptote by degrees
What is lim x to infinity of (3x^2 + x)/(2x^2 - 5), the horizontal asymptote of the graph?
10. Power law for limits
Given lim x to a of f equals L, what does the power law say about lim x to a of f(x)^n for a positive integer n?
11. Product law value
lim x to 2 of f equals 4 and lim x to 2 of g equals 3. By the product law, what number is lim x to 2 of f times g? Answer with a number.
12. Substitute a polynomial
What is lim x to 1 of (x^2 + 3x)? Answer with a number.
13. Leading term dominance
What is lim x to infinity of 5/x? Answer with a number.
14. Degree comparison
What is lim x to infinity of (2x^3 + 1)/(x^3 - x)? Answer with a number.
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