Question

Ction f(x)=e^(-x)-3. Determine tion of the horizontal asymptote.

Answer 1

The horizontal asymptote of the function
`f(x) = e-x - 3 is y = -3.`

Explanation;

In order to determine the horizontal asymptote of the given function
`\(f(x) = e^(-x) - 3\)` we need to analyze the behavior of the function as (x) approaches positive and negative infinity. The function
`(e^(-x)\)`approaches zero as (x) goes to infinity, and since it is subtracted by 3, the overall function approaches
`\(-3\) as \(x\)`tends towards positive infinity. Mathematically, this can be expressed as:

`lim (e^(-x) - 3) = -3`

Similarly as (x) goes to negative infinity
`\(e^(-x)\)` approaches infinity, but againsubtracting 3 from it results in the function approaching
`\(-3\):`

`\[ \lim_{{x \to -\infty}} (e^(-x) - 3) = -3 \]`

Therefore, the horizontal asymptote of the function
`\(f(x) = e^(-x) - 3\) is \(y = -3\)`. This means that as (x) becomes extremely large in either the positive or negative direction, the values of the function (
`f(x)\`) will get arbitrarily close to (-3). The presence of the exponential
`term \(e^(-x)\`) ensures that the function approaches but never reaches (-3)resulting in a horizontal asymptote at (y = -3).