MHB How Can You Solve This Limit Problem Without L'Hospital's Rule?

[bennyzadir]

I would really appreciate if you could help me solving this limit problem!

Determine the limit without using L'Hospital's rule!

$$ \lim_{x\to -2} \sin(\frac{\pi x}{2})\frac{x^2+1}{x+2} = ?$$

Thank you in advance!

 

[Nono713]

Do you know or have proved (or in general are allowed to use) the limit below?
$$\lim_{u \to 0} \frac{\sin{u}}{u} = 1$$
If so, apply partial fraction decomposition to the rational function (the quotient of the two polynomials; that's not actually a polynomial) and expand the limit. You can then evaluate two of the three terms, and for the remaining term try and get it in the form of the limit above for some $u$ to determine the limit of your expression. Good luck!

 

[chisigma]

I would really appreciate if you could help me solving this limit problem!

Determine the limit without using L'Hospital's rule!

$$ \lim_{x\to -2} \sin(\frac{\pi x}{2})\frac{x^2+1}{x+2} = ?$$

Thank you in advance!

Take into account the simple identity...

$\displaystyle \sin \frac{\pi\ x}{2} = - \sin (\frac{\pi\ x}{2} + \pi) = - \sin [\frac{\pi}{2}\ (x+ 2)]\ (1)$

Kind regards

$\chi$ $\sigma$