Limits - find parameters α , β

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February 4th 2013, 12:31 PM
theintervurt

Limits - find parameters α , β

I am given this limit and am asked to find parameters α and β

$\lim _{x \mapsto 1 } \frac{x^2 + \alpha x + \beta}{x^2 - 1} = 2$
February 4th 2013, 01:03 PM
Plato

Re: Limits - find parameters α , β

Quote:

Originally Posted by theintervurt

I am given this limit and am asked to find parameters α and β

$\lim _{x \mapsto 1 } \frac{x^2 + \alpha x + \beta}{x^2 - 1} = 2$

You want $x^2 + \alpha x + \beta=(x-1)(x+c)$ so
$c-1=\alpha\\-c=\beta\\1+c=4$ .
February 4th 2013, 09:20 PM
theintervurt

Re: Limits - find parameters α , β

Thanks, I do understand the reasoning behind this however I am curious is there another way of going about it without having to introduce a new parameter (c) ?

Also here is another similar case which I can't solve

$\lim _{x\mapsto 0} \frac {\gamma + \delta \sin {x}}{x^2} = 3$

again I have to find γ and δ however I haven't been able to figure out how to proceed
February 5th 2013, 06:23 AM
Plato

Re: Limits - find parameters α , β

Quote:

Originally Posted by theintervurt

Thanks, I do understand the reasoning behind this however I am curious is there another way of going about it without having to introduce a new parameter (c) ?
Also here is another similar case which I can't solve
$\lim _{x\mapsto 0} \frac {\gamma + \delta \sin {x}}{x^2} = 3$

First, Never, never continue an existing thread with a new question.
With a new question always start a new thread.

Now, as written $\lim _{x\mapsto 0} \frac {\gamma + \delta \sin {x}}{x^2} = 3$ has no solution.

If it were $\lim _{x\mapsto 0} \frac {\gamma x + \delta \sin {x}}{x^2} = 3$ there is a solution.

Now, I am absolutely not a fan of using l'Hopital's rule for limits, but in this case it is useful.