Need help with reading a text from Stewart's Calculus

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tan(x)=sin(x)/cos(x)
Since if cos(x)=0 then sin(x)=1 (which happens when x=𝛑/2+n𝛑) then you’re dividing 1 by 0 which gives a vertical asymptote
Therefore, all vertical asymptotes of tan(x) could be expressed as x=𝛑/2+n𝛑 where n is an integer

hope this helps

to be more precise, the limit at 𝛑/2+n𝛑 does not exist because on one side it tends to positive infinity and on the other it tende to negative infinity

this is what creates the asymptote

[flat bay]

if on one side it goes to infinity and on the other negative infinity

then yes

and the numerator has to be non-zero

[flat bay]

this is the graph for 1/x

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here it is not defined and there is a vertical asymptote

this is the graph of 1/x^2

here it only tends to positive infinity

it is defined