Solving Limit Without L'Hospital's Rule

[chislam]

I was working out a limit problem and even though I solved it using L'hospital's rule, I do not know how to solve it another way which was probably the intended way. Here's the limit:

\lim_{x\rightarrow2} \frac{x^4 - x^3 + x^2 - 4x - 4}{2x^2 - 5x + 2}

So I used L'hospital's rule and got 20/3 but I want to know another method.

Thanks

 

[Dick]

The numerator and denominator have a common factor of (x-2) that is causing it to have a 0/0 form. If you cancel out that common factor, you can just put x=2.

 

[chislam]

Yeah I figured that was the case, but I wasn't able to factor it out in the numerator.

 

[Dick]

Just do polynomial division of (x-2) into the numerator. You know you'll get zero remainder, right?

 

[chislam]

Doh! Lol, I completely forgot about synthetic division.