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Limits
Limits A limit is a special value a function approaches as its input approaches a specific value. It tells us the function's value "sits" at that point, eve...
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Limits A limit is a special value a function approaches as its input approaches a specific value. It tells us the function's value "sits" at that point, eve...
Limits
A limit is a special value a function approaches as its input approaches a specific value. It tells us the function's value "sits" at that point, even though it may be undefined at that very point.
Formal Definition:
Let f(x) be a function defined on an open interval (a, b). The limit of f(x) as x approaches a (from the left and right sides) is L if, for any given ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever x is in (a, b) and |x - a| < δ.
Examples:
lim_(x -> 0) x^2 = 0 (since 0^2 = 0)
lim_(x -> 100) 1/x = 0 (since 100/100 = 1)
lim_(x -> 2) (x + 1) = 3 (since 2 + 1 = 3)
Key Points:
Limits can be both right and left limits.
Limits of functions may be finite, infinity, or undefined.
Limits can be calculated using various methods, such as direct substitution, factoring, and the limit laws.
Limits have important applications in many fields, including physics, economics, and engineering