LIMITS OF FUNCTIONS
Maths.CLIL
- Limit of a function
- Asymptotes
- Continuity
- Discontinuity
LIMITS OF FUNCTIONS
Maths.CLIL
- Limit of a function
- Asymptotes
- Continuity
- Discontinuity
Limit of a function
Let f(x) be a function defined in a neighbourhood of x0 (except possibly for x0 itself). The limit of f(x) as x approaches x0 is l if, for any ε > 0, there can be found a neighbourhood I of x0 such that | f(x) - l | < ε for every x in I.
Asymptotes
An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.
- The line x = a is a vertical asymptote of the graph of the function y = f(x) if limx→a f(x) = ∞.
- The horizontal line y = c is a horizontal asymptote of the function y = f(x) if limx→∞ f(x) = c.
- The oblique asymptote is y = mx + q if limx→∞ f(x)/x = m and limx→∞ [f(x) - mx] = q.
Continuity
The function f is continuous at a limit point x0 of its domain if the limit of f(x) as x approaches x0 exists and is equal to f(x0): limx→x0 f(x) = f(x0).
This means that three conditions have to be satisfied:
- f has to be defined at x0;
- the limit has to exist;
- the value of the limit must equal f(x0).
We say that a function is continuous on an interval [a, b] if it is defined on that interval and is continuous at every point of that interval. At the endpoints, we only use the appropriate one-sided limit.
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Maths.CLIL
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![Maths.CLIL analisi matematica, limiti notevoli]()
Maths.CLIL analisi matematica, limiti notevoli
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![Maths.CLIL, geometria analitica, la circonferenza]()
Maths.CLIL, geometria analitica, la circonferenza
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![Maths.CLIL analisi matematica, teoremi sui limiti]()
Maths.CLIL analisi matematica, teoremi sui limiti
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