Assume for some real number L and c(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\displaystyle\lim_{x\rightarrow c} f(x) = ∞[/itex] and [itex]\displaystyle\lim_{x\rightarrow c} g(x) = L[/itex]

We must prove

[itex]\displaystyle\lim_{x\rightarrow c} [f(x) + g(x)] = ∞[/itex]

Let [itex]M > 0[/itex]. We know [itex]\displaystyle\lim_{x\rightarrow c} f(x) = ∞[/itex]. Thus,

there exists [itex]δ_1>0[/itex] such that if [itex]0 < |x - c| < δ_1[/itex] we have,

[itex]f(x) > M - L + 1[/itex].

Also, we know [itex]\displaystyle\lim_{x\rightarrow c} g(x) = L[/itex]. Thus,

there exists [itex]δ_2 > 0[/itex] such that if [itex]0 < |x - c| < δ_2[/itex] we have,

[itex]0 < |g(x) - L| < 1 →→ -1 < g(x) - L < 1 →→ L - 1 < g(x) < L + 1[/itex]

Let [itex]δ = min(δ_1,δ_2)[/itex]. And so if [itex]0 < |x - c| < δ[/itex] we will have both,

[itex]f(x) > M - L + 1[/itex] and [itex]g(x) > L - 1[/itex]

Thus,

[itex] f(x) + g(x) > M - L + 1 + L - 1 = M [/itex]

Now what confuses me is how the proof can get to the point

[itex]f(x) > M - L + 1[/itex]

without the assumption that [itex] L > 0 [/itex].

Afterall following from the definition of an infinite limit

[itex]M - L + 1 > 0[/itex] and [itex] M > 0 [/itex]. Right?

Im sure I must be confusing something here. Any help would be appreciated.

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# Confusion regarding a proof for an infinite limit property.

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