# How would you solve this limit?

How can you solve this limit without using the aid of a graphing calculator?

lim x-> 7 (x^2−15x+56)/ sin(x-7)

I can figure it out using a graphing calculator, or by inputting numbers really close to 7, but how can I solve this algebraically without use of a calculator?

Hint: Note that $$\frac{x^2-15x+56}{\sin(x-7)}=(x-8)\frac{x-7}{\sin(x-7)}$$ and use the limit $$\lim_{u\to0}\frac{\sin(u)}{u}=1$$
$$\frac{x^2-15x+56}{\sin(x-7)}= \frac{x-7}{\sin(x-7)} (x-8)$$
Put: $x-7=h$ then the limit is:
$$\ell= \lim_{h \to 0} \frac{h-1}{\frac{\sin h}{h}}=-1$$