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: .$\displaystyle y \:=\:\frac{x^23x+4}{x^2+2x1}$
and I found that there is a horizontal asymptote at $\displaystyle y=1.$
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 $\displaystyle f(x)$ for large values of $\displaystyle x.$
That is, as we go to the far right (or far left), the graph approaches $\displaystyle y = 1.$
However, locally, the function can have $\displaystyle y = 1.$
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.