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## Main Question or Discussion Point

Suppose you have one limit

[tex]

lim_{x\rightarrow \ 0}(cos(x)/x) = \infty

[/tex]

and a second limit

[tex]

lim_{x\rightarrow \ \infty}(x) = \infty

[/tex]

What is the first limit subtracted by the second? Is it simply indeterminate because its inf - inf?

One friend suggested I assume x=cos(y)/y for the second limit then change the second limit to look as follows:

[tex]

lim_{x\rightarrow \ \infty}(x) =? lim_{y\rightarrow \ 0}(cos(y)/y)

[/tex]

Then can I say:

[tex]lim_{x\rightarrow \ 0}(cos(x)/x) - lim_{y\rightarrow \ 0}(cos(y)/y) =? 0 [/tex] ?

[tex]

lim_{x\rightarrow \ 0}(cos(x)/x) = \infty

[/tex]

and a second limit

[tex]

lim_{x\rightarrow \ \infty}(x) = \infty

[/tex]

What is the first limit subtracted by the second? Is it simply indeterminate because its inf - inf?

One friend suggested I assume x=cos(y)/y for the second limit then change the second limit to look as follows:

[tex]

lim_{x\rightarrow \ \infty}(x) =? lim_{y\rightarrow \ 0}(cos(y)/y)

[/tex]

Then can I say:

[tex]lim_{x\rightarrow \ 0}(cos(x)/x) - lim_{y\rightarrow \ 0}(cos(y)/y) =? 0 [/tex] ?