[Intensive lectures] Function limit in advanced mathematics: the limit when the independent variable tends to infinity
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Column: Advanced Mathematics
Table of contents
1. The concept that the function limit independent variable tends to infinity
2. Judgment method for function limit independent variable tends to infinity
3. The property that the function limit independent variable tends to infinity
Necessary memory knowledge points
preface
In advanced mathematics, the limit of a function is one of the important concepts to study the tendency and properties of a function at a certain point or at infinity. The limit of a function is a special case of our concern when the independent variable tends to infinity. Understanding the limit of a function as the independent variable tends to infinity helps us study the growth, trend, and asymptote properties of the function. This article will explain in detail the concept, judgment method and related properties when the independent variable tends to infinity in the function limit.
1. The concept that the function limit independent variable tends to infinity
When the independent variable x tends to infinity, the limit of the function f(x) expresses the tendency and properties of the function when the independent variable approaches infinity. We are concerned with the limiting value of the function f(x) as x tends to positive infinity or negative infinity.
2. Judgment method for function limit independent variable tends to infinity
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Basic definition: The limit of the function f(x) when the independent variable tends to positive infinity is L, which is recorded as lim(x→∞) f(x) = L, if for any given ε (ε > 0), there exists A positive number M such that |f(x) - L| < ε holds for x > M.
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The nature of the limit:
- If lim(x→∞) f(x) = L, then lim(x→∞) (c * f(x)) = c * L, where c is a constant.
- If lim(x→∞) f(x) = L1, lim(x→∞) g(x) = L2, then lim(x→∞) (f(x) + g(x)) = L1 + L2.
- If lim(x→∞) f(x) = L1, lim(x→∞) g(x) = L2, then lim(x→∞) (f(x) * g(x)) = L1 * L2.
- If lim(x→∞) f(x) = L1, lim(x→∞) g(x) = L2 (where L2 ≠ 0), then lim(x→∞) (f(x) / g(x) ) = L1 / L2.
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Asymptote:
- If lim(x→∞) f(x) = L (L is a finite number), then f(x) = L is the horizontal asymptote of the function f(x).
- If lim(x→∞) f(x) = ±∞, then x = a is the vertical asymptote of the function f(x).
3. The property that the function limit independent variable tends to infinity
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The limit of a constant function: The limit of a constant function is itself, that is, lim(x→∞) c = c, where c is a constant.
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Limit of polynomial function: For polynomial function P(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_1 * x + a_0 (where a_i is a constant), When x tends to infinity, its limit is lim(x→∞) P(x) = lim(x→∞) a_n * x^n = ±∞, depending on the sign of the coefficient a_n of the highest order term.
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Limits of exponential and logarithmic functions: As x tends to positive infinity, the limit of the exponential function f(x) = a^x (where a > 1) is lim(x→∞) a^x = +∞; naturally The limit of the logarithmic function ln(x) is lim(x→∞) ln(x) = +∞.
Necessary memory knowledge points
Examples (used to familiarize yourself with the function limit: the limit when the independent variable tends to infinity)
Example 1
Example 2
Example 3
Example 4
in conclusion
Function limit is an important concept to study the trend and properties of a function at a certain point or at infinity. We are concerned with the extreme value of the function as the independent variable tends to infinity. Through the basic definition, the properties of the limit and the properties of the asymptote, we can determine and calculate the limit of the function when the independent variable tends to infinity. Understanding the concept and nature of the limit independent variable of a function tends to infinity helps us analyze the growth trend and asymptote of the function and solve optimization and extreme value problems in practical problems.