Calculus Volume 1 · 免费预览
洛必达法则
0/0 和 ∞/∞ 这两种未定式可用洛必达法则处理，该法则把商替换为导数之比，从而解决那些难以用代数方法化简的极限，并判定函数增长速度的层级。
1. The rule and its justification
The rule and its justification
2. Standard applications
Standard applications
3. The growth hierarchy proved
The growth hierarchy proved
4. Other indeterminate forms
Other indeterminate forms
5. What L'Hôpital differentiates
L'Hôpital's rule evaluates lim f/g for 0/0 by taking which expression?
6. One application
What is lim x to 0 of (e^x - 1)/x, a 0/0 form, after one application of L'Hôpital's rule?
7. Repeated application
lim x to 0 of (1 - cos x)/x^2 requires two applications of L'Hôpital's rule. What is the final value?
8. Product form reduction
The limit lim x to 0 from the right of x ln x is the indeterminate product 0 times infinity. Which rewriting puts it into a L'Hôpital form?
9. Exponential dominance
What does L'Hôpital's rule show about lim x to infinity of e^x over x^n for positive n?
10. When the rule does not apply
For which of these limits is L'Hopital's rule NOT applicable, even though it may look tempting?
11. Single application limit
lim x to 0 of sin(3x)/x is 0/0. After one application of L'Hôpital's rule the limit becomes lim of 3 cos(3x) over 1. What number is this limit? Answer with a number.
12. Log over power
lim x to infinity of (ln x)/x^2 is infinity/infinity. After one application the numerator becomes 1/x. What does the denominator become? Answer like 2x.
13. Number of applications for x cubed
To evaluate lim x to infinity of e^x/x^3 by L'Hôpital's rule, the denominator must be differentiated until constant. How many applications are needed? Answer with a number.
14. Value of x to the x at zero
lim x to 0 from the right of x ln x equals 0, so lim x to 0 from the right of x^x equals e to what power? Answer with a number.