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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?

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

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Hint:

$$\frac{x^2-15x+56}{\sin(x-7)}= \frac{x-7}{\sin(x-7)} (x-8)$$

Put: $x-7=h$ then the limit is:

$$\ell= \lim_{h \to 0} \frac{h-1}{\frac{\sin h}{h}}=-1$$

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