limit with greatest integer function How can I evaluate this limit?
$$\lim_{x\to-7} \frac{[x]^2+15[x]+56}{\sin(x+7)\sin(x+8)}$$ where $[x]$ denotes the greatest integer less than or equal to $x$
I could easily factor the numerator. But I cannot apply any of the standard limits due to $[x]$.

The way the question is phrased makes me wonder if you've understood the question, either the nature of limits or the nature of the greatest-integer function.  Your final sentence makes me suspect the latter.
If $x<-7$ then $\lfloor x\rfloor=-8$.
If $x>-7$ then $\lfloor x\rfloor= -7$.
It you need that explained to you you then you haven't understood the greatest-integer function and that would be what you need to look at.
In one case the numerator is $(-8)^2+15(-8)+56=0$ and in the other case it is $(-7)^2 + 15(-7)+56 = 0$.  Either way the fraction is equal to $0$ in small neighboorhoods about $-7$ unless the denominator is $0$.  And in small neighborhoods about $-7$, the denominator is $0$ only if $x=-7$, so essentially your asking for $\lim\limits_{x\to-7} 0$.
