Calculus Volume 1 · 免费预览
无穷远处的极限与渐近线
端点行为通过无穷远处的极限来计算，所得极限给出水平渐近线；在有限点附近的无界行为给出垂直渐近线；斜渐近线则刻画次数相差一的有理函数。
1. Horizontal asymptotes from limits at infinity
Horizontal asymptotes from limits at infinity
2. Vertical asymptotes
Vertical asymptotes
3. Oblique asymptotes
Oblique asymptotes
4. Transcendental end behavior
Transcendental end behavior
5. Horizontal asymptote rule
The rational function (3x^2 + x)/(2x^2 - 5) has what horizontal asymptote?
6. Hole versus asymptote
The function (x^2 - 1)/(x^2 - 1) would reduce to 1 everywhere except at x = plus or minus 1. What does the graph have at those points?
7. One-sided signs at a vertical asymptote
Near x = 1, the function 1/(x - 1) has what one-sided limit behavior?
8. Oblique asymptote by division
Polynomial division of (x^2 + 2x)/(x - 1) gives x + 3 plus a remainder term. What line is the oblique asymptote?
9. End behavior of the exponential
Which pair correctly states the two end behaviors of f(x) = e^x?
10. Crossing a horizontal asymptote
The function sin(x)/x approaches 0 as x approaches infinity, so y = 0 is a horizontal asymptote. What else is true of this function?
11. Limit at infinity of 1 over x
What is lim x to infinity of 1/x? Answer with a number.
12. Vertical asymptote location
The function 1/(x - 2)^2 has an unbounded blow-up at one point. At what x does the vertical asymptote occur? Answer with a number.
13. Degree excess for oblique
An oblique asymptote occurs when the numerator's degree exceeds the denominator's by exactly one. If the denominator has degree 2, what numerator degree gives an oblique asymptote? Answer with a number.
14. Horizontal asymptote of arctan
As x approaches infinity, arctan x approaches pi/2. What horizontal asymptote does the graph have on the right? Answer like y = pi/2.