_{x->0}1/x

^{2 }does not exist (DNE) as a finite number;

_{x->0}1/x

^{2}= ∞ really comes from a definition. And many courses say that all limits -> ∞ DNE.

**Definition**We say that a function f(x) diverges to infinity, denoted by f(x) -> ∞ as x -> a,

|f(x)| = | 1/x | = 1/|x| > ε whenever | x - 0 | = |x| < δ = 1/ε.

**Proposition**The function f(x) -> ∞ as x -> a if and only if the function 1/f(x) -> 0 as x -> a.

_{x->0}1/x

^{2}= (lim

_{ x->0}1) / (lim

_{ x->0}x

^{2})

_{x->a}x = a and lim

_{x->0}f(x

^{2}) = (lim

_{x->0}f(x))

^{2}

_{x->0}x)

^{2}= 1 / (0)

^{2}= undefined

^{2}

^{2}-> 0 (by substitution, or more formally.

^{2}-> 0 Then f(x) = 1/x

^{2}-> ∞ or DNE

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