# Please get me started on showing that the following limit exists

• kingstrick
In summary, when evaluating the limit (x→∞) lim ((x+2)/√x) where (x > 0), the numerator is always larger than the denominator, indicating that the limit does not exist in the traditional sense. However, we can say that the limit is ∞, as the function gets larger without bound as x approaches infinity.
kingstrick

## Homework Statement

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

## 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.

kingstrick said:

## Homework Statement

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

## 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.

Mark44 said:
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

kingstrick said:
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

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

kingstrick said:
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.

Mark44 said:
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.

kingstrick said:
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 -(1x).

Thank you. I think i understand now.

## 1. What does it mean for a limit to exist?

A limit exists if the value of a function approaches a specific number as the input approaches a certain value.

## 2. How do I know if a limit exists?

A limit exists if the left-sided limit and right-sided limit are equal at the specific value of the input.

## 3. What is the process of proving that a limit exists?

To prove that a limit exists, you must first determine if the limit is a one-sided or two-sided limit. Then, you can use the definition of a limit to show that the left-sided limit and right-sided limit are equal.

## 4. Can a limit exist but not be defined at a certain point?

Yes, it is possible for a limit to exist but for the function to not be defined at that point. This can happen if there is a hole or a jump in the graph of the function at that point.

## 5. How can I use the limit definition to show that a limit exists?

The limit definition states that for a limit to exist, the absolute value of the difference between the function value and the limit value must be smaller than a certain value, for all values of the input that are within a specific distance from the limit value. By showing that this condition is satisfied, you can prove that the limit exists.

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