Limits
Prerequisites Math glossary
Definitions
If \(x\) approaches closely to some value \(a\) but not equals to it, we denote this as \(x \to a\). When it approaches only by values that are more than \(a\) we say that it does it from the right \(x \to a^+\). For less than \(a\) values it approaches from the left \(x \to a^-\).
If \(f(x) \to R\) when \(x \to a^+\) there is a right-hand limit. \[\lim_{x \to a^+} f(x) = R\] If \(f(x) \to L\) when \(x \to a^-\) there is a left-hand limit. \[\lim_{x \to a^-} f(x) = L\] If \(R = L\) then \(f(x)\) has a limit. \[\lim_{x \to a} f(x) = L = R\]
Formal definition
If \(\forall \epsilon>0 \; \exists \delta>0\) such that if \(0<|x - \delta|<a\) then \(|f(x) - L|<\epsilon\)
No limit. Left-hand != Right-hand
Limit exists but function is discontinuous in point
Limit Laws \[\lim (f\pm g) = \lim f \pm \lim g\] \[\lim (f*g) = \lim f * \lim g\] \[\lim (f/g) = \lim f / \lim g, \lim g \neq 0\] \[\lim (f/g) = DNE, \lim f \neq 0, \lim g = 0\] \[\lim (f/g) = ?, \lim f = 0, \lim g = 0\]
Function is continuous in the point \(a\) when \(\lim_{x \to a} f(x) = f(a)\). Function is continuous if it’s continuous for any value of its argument. if \(f\) is continuous in \(b\) \[\lim_{x \to a} f(g(x)) = f(b), \lim_{x \to a} g(x) = b\] Intermediate Value Theorem. If \(f\) is a function which is continuous on the interval \([a,b]\) (right-continuous in \(a\), left-continuous in \(b\), all points between \(a\) and \(b\)), and \(M\) lies between the values of \(f(a)\) and \(f(b)\), then there is at least one point \(c\) between \(a\) and \(b\) such that \(f(c)=M\).
Limits