Question

Determine whether the given limit leads to a determinate or indeterminate form. HINT [See Example 2.]

lim

x→−[infinity] 2/−x + 3


determinate form

indeterminate form


Evaluate the limit if it exists. (If you need to use or –, enter INFINITY or –INFINITY, respectively. If an answer does not exist, enter DNE.)



If the limit does not exist, say why. (If the limit does exist, so state.)

Answer 1

The limit leads to a determinate form.

`\lim_(x \to \infty) (2)/(-x+3) = 0`

Explanation:

The following are indeterminate forms.

`(0)/(0) \ and \ (\infty)/(\infty)`

Given the limit of a function
`\lim_(x \to \infty) (2)/(-x+3)`, to show if the given limit is determinate or indeterminate form, we will need to substitute the value of -
`\infty` into the function as shown,

`\lim_(x \to \infty) (2)/(-x+3)\\= (2)/(-(-\infty)+3)\\= (2)/(\infty+3)\\= (2)/(\infty)\\\\Generally, \ (a)/(\infty) =0`

where a is any constant, therefore
`(2)/(\infty) = 0`

Since we are able to get a finite value i.e 0, this shows that the limit does exist and leads to a determinate form