Find $\lim_{x \rightarrow 1} \frac{\sqrt{2x^2-1}-1}{x^3-1}$ without using L'Hôpital's rule

Question

I tried:

$$\lim_{x \rightarrow 1} \frac{\sqrt{2x^2-1}-1}{x^3-1} = \\ \lim_{x \rightarrow 1} \frac{\sqrt{2x^2-1}-1}{x^3-1} \cdot \frac{\sqrt{2x^2-1}+1}{\sqrt{2x^2-1}+1}\\ = \lim_{x \rightarrow 1} \frac{4x^4-4x^2}{(x^3-1)(\sqrt{2x^2-1}+1)} = ???$$

My questions:

1. What do I do next? Am I doing it correctly so far?
2. How do you know which method to use when simplifying expressions? How do you know whether if you have to rationalize the denominator or the numerator or both, divide the numerator and the denominator by one or the other, factorize, use more than just one method, etc...? I find myself wasting a lot of time moving from one method to the other never really knowing why I am using a certain method or whether if it works or not. Do you know any tricks or have any advice for me?

Answer 1

There is an error in the third line which is overlooked.

$$\lim_{x \rightarrow 1} \frac{\sqrt{2x^2-1}-1}{x^3-1} = \\ \lim_{x \rightarrow 1} \frac{\sqrt{2x^2-1}-1}{x^3-1} \cdot \frac{\sqrt{2x^2-1}+1}{\sqrt{2x^2-1}+1}\\ = \lim_{x \rightarrow 1} \frac{2x^2-2}{(x^3-1)(\sqrt{2x^2-1}+1)} $$

$$=\lim_{x \rightarrow 1} \frac{2(x-1)(x+1)}{(x-1)(x^2+x+1)(\sqrt{2x^2-1}+1)} $$

$$=\lim_{x \rightarrow 1} \frac{2(x+1)}{(x^2+x+1)(\sqrt{2x^2-1}+1)} = \frac {2}{3}$$

Answer 2

HINT: use that $$x^3-1=(x-1)(x^2+x+1)$$ and $$4x^4-4x^2=4x^2(x-1)(x+1)$$ can you finish now? and use that $$2x^2-1-1=2x^2-2=2(x^2-1)=2(x-1)(x+1)$$

Answer 3

1) You made a small error:

$$\frac{(\sqrt{2x^2-1}-1)(\sqrt{2x^2-1}+1)}{(x^3-1)(\sqrt{2x^2-1}+1)}=\frac{(2x^2-1)-1^2}{(x-1)(x^2+x+1)(\sqrt{2x^2-1}+1)}$$

Note the numerator should be the difference of two squares. Factor out $(x-1)$ from the numerator, and then you may now directly substitute $x=1$.

2) In general, try to spot out which parts of the limit cause the $0/0$ indeterminate form part. Those clearly are factorable, so remove the right factors from them.

Answer 4

Your last step seems incorrect because you didn't multiply the two numerators correctly.

Continue your calculation as follows. Your limit is $$\lim_{x \to 1} \frac{2x^2 - 2}{(x^3 - 1)(\sqrt{2x^2 - 1} + 1)} = \lim_{x \to 1} \frac{2}{\sqrt{2x^2 - 1} + 1} \cdot \lim_{x \to 1}\frac{x^2 - 1}{x^3 - 1} = 1 \cdot \lim_{x \to 1}\frac{x^2 - 1}{x^3 - 1}.$$

Before going any further, let me point out a couple of things.

First, when you're trying to calculate a limit of a quantity expressed as a product or quotient of several factors, any factors that have a finite, nonzero limit are easy to deal with: just split them off as we've done above.

Second, the method of rationalization is most often applied when you have a complicated term $A \pm B$ whose limit is zero or infinity, or unknown, but the expression $A^2 - B^2$ is simpler to deal with. This is useful when the "conjugate" expression $A \mp B$ has a finite nonzero limit.

In this case, the expression $A - B = \sqrt{2x^2 - 1} - 1$ had a zero limit and was therefore difficult to deal with. However, the conjugate term $A + B = \sqrt{2x^2 - 1} + 1$ had a limit of $2\ne 0, \infty$, and the product $(A-B)(A + B) = A^2 - B^2 = 2x^2 - 2$ is simple in form.

To continue the calculation, there are two methods, depending on whether it's easy for you to factor the numerator and denominator.

If it is easy to factor, do this: $$\lim_{x \to 1} \frac{x^2 - 1}{x^3 - 1} = \lim_{x \to 1} \frac{(x-1)(x +1)}{(x - 1)(x^2 + x + 1)} = \lim_{x \to 1} \frac{x + 1}{x^2 + x + 1} = \frac{2}{3}$$

Otherwise, make the substitution $x = 1 + h$, where $h \to 0$: $$\lim_{x \to 1} \frac{x^2 - 1}{x^3 - 1} = \lim_{h \to 0} \frac{(1 + h)^2 - 1}{(1 + h)^3 - 1} = \lim_{h \to 0} \frac{2h + h^2}{3h + 3h^2 + h^3} = \lim_{h \to 0} \frac{h(2 + h)}{h(3 + 3h + h^2)} = \lim_{h \to 0} \frac{2 + h}{3 + 3h + h^2} = \frac{2}{3}$$

Answer 5

Since $x^3-1$ is a polynomial that is $0$ at $x=1,$ we can factor an $(x-1)$ out: $x^3-1 = (x-1)(x^2+x+1).$ Our expression thus equals

$$\tag 1 \frac{\sqrt {2x^2-1} - 1}{x-1}\frac{1}{x^2 + x + 1}.$$

Let $f(x) = \sqrt {2x^2-1}.$ We can write $(1)$ as

$$ \tag 2 \frac{f(x) - f(1)}{x-1}\frac{1}{x^2 + x + 1}.$$

As $x\to 1,$ the first fraction in $(2) \to f'(1)$ by definition of the derivative. The second fraction $\to 1/3.$ The desired limit is thus $f'(1)/3,$ which is a simple computation. (And no L'Hopital was used.)