Hello, dimpleyy!
A common misunderstanding . . .
(I went through it myself many years ago.)
For my project, I have to pick an equation and sketch it.
I picked: .
and I found that there is a horizontal asymptote at
But when I plug in 1 in to the equation there is a point at (1,1).
How does that happen?
A horizontal asymptote indicates the behavior of for large values of
That is, as we go to the far right (or far left), the graph approaches
However, locally, the function can have
The graph might look like this . . . Code:

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     +o               

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You see, the graph can cross the horizontal asymptote.
. . The asymptote takes over on extreme values only.