Question

How would i answer and what would be the answer?

Answer 1

Given:

`f(x)=-(1)/(x-2)+3`

We need to find asymptotes of the given function.

Recall that a horizontal asymptote is a horizontal line, y=a, that has the property that either:

`\lim _(x\to\infty)f(x)=a`

or

`\lim _(x\to-\infty)f(x)=a`

Taking the limit of the given function, we get

`\lim _(x\to\infty)f(x)=\lim _(x\to\infty)(-(1)/(x-2)+3)`

`\lim _(x\to\infty)f(x)=0+3`

`\lim _(x\to\infty)f(x)=3`

The horizontal asymptote is y=3.

Recall that a vertical asymptote is a vertical line, x=a, that has the property that either:

`\lim _(x\to a^-)f(x)=\pm\infty`

or

`\lim _(x\to a^+)f(x)=\pm\infty`

Taking limit to the given function, we get

`\lim _(x\to a^+)f(x)=\lim _(x\to a^+)(-(1)/(x-2)+3)`

If we take a=2, the limit will be infinity.

`\lim _(x\to2^+)f(x)=\lim _(x\to2^+)(-(1)/(x-2)+3)=\infty`

Hence the vertical asymptotes x=2.

From these two asymptotes, we can say that the first option or third option would be the graph.

Consider the point (3,2) from the first option graph.

Substitute x=3 and f(3)=2 in the given function, we get

`2=-(1)/(3-2)+3`

`2=-1+3`
`2=2`

The point (3,2) satisfies the given function.

Hence the graph of the function is