Calculus Volume 1 · 免费预览
函数的极限
函数 f(x) 在 x 趋近 a 时的极限，是其输出值所趋近的数值；即使函数在该点无定义，这个极限也可能存在。极限可以从表格和图像中读出，并有单侧极限和无穷极限等形式。
1. The idea of approach
The idea of approach
2. Estimating from tables and graphs
Estimating from tables and graphs
3. One-sided limits
One-sided limits
4. Limits that fail to exist
Limits that fail to exist
5. Limit at a hole
The function f(x) = (x^2 - 4)/(x - 2) is undefined at x = 2. What is the limit of f(x) as x approaches 2?
6. One-sided agreement
A function has left-hand limit 3 and right-hand limit 5 as x approaches 2. What is the two-sided limit at 2?
7. Table estimation
For f(x) = (sin x)/x near 0, a table gives f(0.1) about 0.9983, f(0.01) about 0.99998, and f(0.001) about 0.9999998, with matching values from the left. What limit do the values suggest?
8. Infinite limit versus limit at infinity
Which statement correctly distinguishes the two notations lim x to 2 of f equals infinity and lim x to infinity of f equals 2?
9. Oscillating nonexistence
Why does the limit of sin(1/x) as x approaches 0 fail to exist?
10. Limit of a constant function
The function f(x) = 7 takes the value 7 everywhere. What is the limit of f(x) as x approaches any real number a?
11. One-sided at an endpoint
The square root function is defined only for x at least 0. What is its right-hand limit as x approaches 0? Answer with a number.
12. Sign of an unbounded limit
As x approaches 0 from the right, 1/x grows without bound in the positive direction. What symbol describes this limit? Answer with the single word infinity.
13. Limit of 1 over x at infinity
The outputs of f(x) = 1/x settle toward a single number as x grows without bound. What is that number? Answer with a number.
14. Two-sided agreement value
A function has left-hand limit 7 and right-hand limit 7 at x = 3. What is its two-sided limit there? Answer with a number.