# Proving Limits at Infinity: Can You Help Me with These Two Limits?

• MHB
• goody1
In summary, the conversation discusses how to prove two limits using the definition of limit. The first limit is \lim_{x\to\infty}\frac{x-1}{x+2} = 1, and the second limit is \lim_{x\to-1}\frac{-1}{(x+1)^2} = -\infty. The steps for proving these limits are explained, with a focus on simplifying expressions and finding appropriate values for $N$ or $\delta$ in order to satisfy the given conditions.
goody1
Hi, can anybody help me with this two limits? I have to prove them by the definition of limit. Thank you in advance.
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goody said:
Hi, can anybody help me with this two limits? I have to prove them by the definition of limit. Thank you in advance.
Hi Goody, and welcome to MHB!

To prove that $$\displaystyle \lim_{x\to\infty}\frac{x-1}{x+2} = 1$$, you have to show that, given $\varepsilon > 0$, you can find $N$ such that $$\displaystyle \left|\frac{x-1}{x+2} - 1\right| < \varepsilon$$ whenever $x>N$.

So, first you should simplify $$\displaystyle \left|\frac{x-1}{x+2} - 1\right|$$. Then you should see how large $x$ has to be in order to make that expression less than $\varepsilon$.

To prove that $$\displaystyle \lim_{x\to-1}\frac{-1}{(x+1)^2} = -\infty$$, you have to show that, given $M$, you can find $\delta>0$ such that $$\displaystyle \frac{-1}{(x+1)^2} < -M$$ whenever $|x+1| < \delta$. That is actually an easier calculation than the first one, so you might want to try that one first.

Hi Opalg! Do you think I got it correct?
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goody said:
Hi Opalg! Do you think I got it correct?
Not quite, although you started correctly. The limit in this case is as $x\to\infty$, so you want to see what happens when $x$ gets large. This means that the inequality $\dfrac3{|x+2|}<\varepsilon$ has to hold for all $x$ greater than $N$ (where you think of $N$ as being a large number).

Write the inequality as $|x+2| > \dfrac3\varepsilon$, and you see that this will be true if $x > \dfrac3\varepsilon -2$. So you can take $N = \dfrac3\varepsilon -2$. More simply, you could take $N = \dfrac3\varepsilon$, which will satisfy the required condition with a bit to spare.

## 1. What is the definition of a limit?

The limit of a function is the value that a function approaches as the input value gets closer and closer to a specific value. It is denoted by the notation lim f(x) as x approaches a.

## 2. How do you prove a limit by definition?

To prove a limit by definition, you must show that for any positive number ε (epsilon), there exists a corresponding positive number δ (delta) such that if the distance between the input value x and the limit value a is less than δ, then the distance between the function value f(x) and the limit L is less than ε.

## 3. What is the importance of proving a limit by definition?

Proving a limit by definition is important because it is the most rigorous way to establish the existence of a limit. It also allows us to understand the behavior of a function near a specific point and make predictions about its behavior.

## 4. What are some common techniques used to prove a limit by definition?

Some common techniques used to prove a limit by definition include algebraic manipulation, the squeeze theorem, and the epsilon-delta definition of a limit.

## 5. Can a limit be proven by definition if the function is not continuous?

Yes, a limit can still be proven by definition even if the function is not continuous. The epsilon-delta definition of a limit does not require the function to be continuous, only that it approaches a specific value as the input value gets closer and closer to a certain value.

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