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例题 1 · A-Level · Series and Limits
Explain the concept of a limit as x approaches a finite value a. Use the notation limx→a f(x) to illustrate your answer.
查看答案
The limit of f(x) as x approaches a is the value that f(x) gets arbitrarily close to as x approaches a (without necessarily equalling a). Formally, lim[x→a] f(x) = L means that for any ε > 0, there exists δ > 0 such that |f(x) − L| < ε whenever 0 < |x − a| < δ. The function need not be defined at x = a for the limit to exist.
例题 2 · A-Level · Series and Limits
Find limx→2 (x² − 4)/(x − 2).
查看答案
lim[x→2] (x²−4)/(x−2) = lim[x→2] ((x−2)(x+2))/(x−2) = lim[x→2] (x+2) = 4
例题 3 · A-Level · Series and Limits
Find limx→0 (sin 3x)/(2x).
查看答案
Rewrite as (sin 3x)/(2x) = (3/2) · (sin 3x)/(3x). As x→0, (sin 3x)/(3x)→1. Therefore limit = 3/2.