Limit (mathematics)
Infinity in limits of functions

It is possible to define the notion of "tending to infinity" in the domain of f {\displaystyle f}, lim x → ∞ f (x) = L. {\displaystyle \lim _{x\rightarrow \infty }f(x)=L.}

In this expression, the infinity is considered to be signed: either + ∞ {\displaystyle +\infty } or − ∞ {\displaystyle -\infty }. The "limit of f as x tends to positive infinity" is defined as follows. It is a real number L {\displaystyle L} such that, given any real ϵ > 0 {\displaystyle \epsilon >0}, there exists an M > 0 {\displaystyle M>0} so that if x > M {\displaystyle x>M}, | f (x) − L | < ϵ {\displaystyle |f(x)-L|<\epsilon }. Equivalently, for any sequence x n → + ∞ {\displaystyle x_{n}\rightarrow +\infty }, we have f (x n) → L {\displaystyle f(x_{n})\rightarrow L}.

It is also possible to define the notion of "tending to infinity" in the value of f {\displaystyle f}, lim x → c f (x) = ∞. {\displaystyle \lim _{x\rightarrow c}f(x)=\infty.}

The definition is given as follows. Given any real number M > 0 {\displaystyle M>0}, there is a δ > 0 {\displaystyle \delta >0} so that for 0 < | x − c | < δ {\displaystyle 0<|x-c|<\delta }, the absolute value of the function | f (x) | > M {\displaystyle |f(x)|>M}. Equivalently, for any sequence x n → c {\displaystyle x_{n}\rightarrow c}, the sequence f (x n) → ∞ {\displaystyle f(x_{n})\rightarrow \infty }.
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