The horizontal asymptote of y = e^x - 1 is y =_________

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An asymptote of a function is a limit the function tends to in either the `x`direction or the `y`direction.

As this is a horizontal asymptote the asymptote is in the `x`direction and is a particular value of `y`.

Look at both ends of the `x`axis, ie when `x`is either very large or very small.

For very large `x`we write `x -> oo`

Since the value of `e` is greater than 1, as `x-> oo` the value of `e^x` also gets very large, that is `e^x -> oo`

and `e^x -1 -> oo` since the magnitude of the -1 term is negligible compared to the size of `e^x`as `x -> oo`.

 

For very small `x`we write `x -> -oo`

Again since the value of `e` is greater than 1, as `x -> -oo` the value of `e^x` gets very close to zero, that is `e^x -> 0`

and `e^x - 1 -> -1` . This time the magnitude of the -1 term is large compared to the size of `e^x` as `x -> -oo` .

Therefore the horizontal asymptote is y = -1

 

 

 

 

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