Please get me started on showing that the following limit exists

[kingstrick]

Homework Statement

Evaluate the limit or show it doesn't exist. (x→∞) lim ((x+2)/√x) where (x > 0)

Homework Equations

The Attempt at a Solution

I know how to solve it if x → c but i don't know how to start it when it goes to infinity. I just need a hint as to how to start the problem.

 

[Mark44]

Homework Statement

Evaluate the limit or show it doesn't exist. (x→∞) lim ((x+2)/√x) where (x > 0)

Homework Equations

The Attempt at a Solution

I know how to solve it if x → c but i don't know how to start it when it goes to infinity. I just need a hint as to how to start the problem.

Divide each term in the numerator by √x, and then take the limit.

 

[kingstrick]

Divide each term in the numerator by √x, and then take the limit.

Mark, thanks for responding,

so after evaluating, i found that the limit does not exists... am i correct?
work:
((x+2)/√x)/1 --- Does not exist since the numerator is always bigger than the denominator
x→∞, x > o

 

[scurty]

Mark, thanks for responding,

so after evaluating, i found that the limit does not exists... am i correct?
work:
((x+2)/√x)/1 --- Does not exist since the numerator is always bigger than the denominator
x→∞, x > o

That's not what Mark meant. You essentially have the exact same equation you started with. Remember that

$\displaystyle\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$.

 

[Mark44]

so after evaluating, i found that the limit does not exists... am i correct?
work:
((x+2)/√x)/1 --- Does not exist since the numerator is always bigger than the denominator
x→∞, x > o

In one sense, the limit doesn't exist, because (x + 2)/√x gets large without bound as x gets large. In that sense, the limit doesn't exist because it is not a finite number. For limits like this, though, we usually say that the limit is ∞.

Also, because x is approaching infinity, you don't need to say that x > 0.

 

[kingstrick]

In one sense, the limit doesn't exist, because (x + 2)/√x gets large without bound as x gets large. In that sense, the limit doesn't exist because it is not a finite number. For limits like this, though, we usually say that the limit is ∞.

Also, because x is approaching infinity, you don't need to say that x > 0.

So when x → ∞, a function will always either go to a finite number, ∞, or -∞...so will a function then always have a limit in this sense? Oh, nevermind, some functions can diverge, like f(x) = -1^x.

 

[Mark44]

So when x → ∞, a function will always either go to a finite number, ∞, or -∞...so will a function then always have a limit in this sense? Oh, nevermind, some functions can diverge, like f(x) = -1^x.

Right, except that $\lim_{x \to \infty}-1^x = -1$

The one you're thinking of is f(x) = (-1)^x. Without parentheses, what you wrote is the same as -(1^x).

 

[kingstrick]

Thank you. I think i understand now.