1. Definition of limit:
Limit is the stable value of function value at a certain point or infinity.
The formal definition is as follows:
If for any given positive number ε, there is a positive number δ, so when 0
lim (x→a) f(x) = L
2. Basic limit formula:
Lim (x→c) k = k, where k is constant.
lim (x→c) x = c .
Lim (x → c) x n = c n, where n is a positive integer.
Forest (x → c) e x = e c.
Lim (x→c) a^x = a^c a c a c, where a is a positive number.
3. Four algorithms of limit:
Sum and difference rule of limit: lim (x→ c) [f (x) g (x)] = lim (x→ c) f (x) lim (x→ c) g (x).
The multiplication rule of limit: lim (x→ c) [f (x) * g (x)] = lim (x→ c) f (x) * lim (x→ c) g (x).
Division rule of limit: lim (x→ c) [f (x)/g (x)] = (lim (x→ c) f (x))/(lim (x→ c) g (x)), provided that lim (x→c) g(x) ≠ 0.
Power law of limit: lim (x → c) [f (x) n] = [lim (x → c) f (x)] n.
4. Infinite and infinitesimal:
The limit is infinite: lim (x→c) f(x) = ∞ or lim (x→c) f(x) = -∞
Limit infinitesimal: lim (x→c) f(x) = 0.
5. Common special restrictions:
lim (x→0) sin(x)/x = 1
lim(x→0)(e^x- 1)/x = 1
lim(x→∞)( 1+ 1/x)^x = e
Lim (x →∞) (1+a/x) x = e a, where a is a constant.
lim(x→0)( 1+x)^ 1/x = e
6. Conditions for the existence of limit:
The limit of a function exists at a certain point, which requires the function to be defined near that point.
The existence of limit is not necessarily equal to the function value at this point.
7. Nature of the restriction:
Boundedness: if lim (x→c) f(x) exists, then f(x) is bounded around x = C.
Title reserved: if lim (x → c) f (x) >; 0 (or; 0 (or
Pinch theorem: If there are two functions g(x) and h(x), which satisfy that g(x) ≤ f(x) ≤ h(x) is near a certain point, and lim (x→c) g(x) = lim (x→c) h(x) = L, then lim (x→c).
Limit is an important concept in higher mathematics. It is not only the foundation of calculus, but also widely used in analysis, engineering and physics. Mastering the definition, basic formula and algorithm of limit, as well as the nature of special limit will help you better understand and solve the mathematical problems related to limit. When learning the limit, practical practice and application are also important. By doing more exercises and exploring limit problems in different situations, you can improve your math skills and problem-solving ability.