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I've just have a mathematics exam and a question was this:

Calculate the limits of $\dfrac{x^2+3x+2}{x^2+2x+1}$ when $x\text{ aproaches }-1$.

I started by dividing it using the polynomial long division. But I always get $\frac{0}{0}$.

How is this limit evaluated?

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@BrianM.Scott Edited. – Garmen1778 May 10 '12 at 18:27
@Gigili You could salvage your deleted answer by looking at one sided limits. L'Hopital is applicable, e.g., when $\lim_{x\rightarrow a^+}{f'(x)\over g'(x)}=\infty$. So here you can show the limit from the left is $-\infty$ and the limit from the right is $\infty$. I think it would be worth while to write this up. – David Mitra May 10 '12 at 19:19
@DavidMitra: Thank you, now I got your point. Perhaps I was sleepy yesterday! – Gigili May 11 '12 at 4:31
up vote 7 down vote accepted

Hint for an alternate method: ${x^2+3x+2\over x^2+2x+1}={(x+2)(x+1)\over(x+1)(x+1)} $. (After cancelling, look at the limits from the left and right.)

Long division will get you the result after a bit more work as well. Doing the division reveals $$ {x^2+3x+2\over x^2+2x+1} =1+{x+1\over x^2+2x+1}. $$ Now, using $x^2+2x+1=(x+1)^2$, write this as $$ 1+{1\over x+1}. $$ And then consider $\lim\limits_{x\rightarrow -1}\Bigl(1+{1\over x+1}\Bigr)$. (I think the first method above is a bit less work.)

Can you handle this now?

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If you divided correctly, you got $$1+\frac{x+1}{x^2+2x+1}\;.\tag{1}$$ You should recognize that the denominator is simply $(x+1)^2$, so $(1)$ is $$1+\frac{x+1}{(x+1)^2}=1+\frac1{x+1}\;.$$ Thus, your limit is $$\lim_{x\to-1}\left(1+\frac1{x+1}\right)=1+\lim_{x\to-1}\frac1{x+1}\;,$$ and you shouldn't have too much trouble seeing what's going on with that last limit.

Had you recognized that the original numerator is $(x+1)(x+2)$, you could have divided out the common factor of $x+1$ and reached this stage somewhat more quickly.

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Another approach is to exploit the fact that it's easier to compute limits of rational functions at $0$. Thus changing variables $\rm\:z = x\!+\!1\:$ shifts $\rm\:x = -1\:$ to $\rm\:z = 0,\:$ so with $\rm\:x = z\!-\!1\:$ we get

$$\rm\ \lim_{z\to 0} \frac{(z\!-\!1)^2+3(z\!-\!1)+2}{(z\!-\!1)^2+2(z\!-\!1)+1}\ =\ \lim_{z\to 0}\frac{z^2 + z}{z^2}\ =\ \lim_{z\to 0}\:\left(1 + \frac{1}z\right)$$

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You can do also this way,

Using l'Hospital rule,

$\lim_{x \to -1} \frac{x^2+3x+2}{x^2+2x+1} \ \ \ \ \ \ (\frac{0}{0}form)$

Applying L'hospital rule,

$\lim_{x \to -1} \frac{2x+3}{2x+2}$

$\lim_{x \to -1} \frac{2x+2+1}{2x+2}$

$\lim_{x \to -1} 1+\frac{1}{2x+2}$

can you do this now?

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