# Find the limits: lim_{xrightarrow2}frac{x^2-7x+10}{x-2}

Question
Limits and continuity
Find the limits:
$$\lim_{x\rightarrow2}\frac{x^2-7x+10}{x-2}$$

2020-11-10
Apply the limits to the function.
$$\lim_{x\rightarrow2}\frac{x^2-7x+10}{x-2}=\frac{2^2-7\times2+10}{2-2}$$
$$=\frac{4-14+10}{0}$$
$$=\frac{0}{0}$$
On applying the limits directly, the function gives $$\frac{0}{0}$$ form. So apply the L'Hospitals rule to the function.
$$\lim_{x\rightarrow2}\frac{f(x)}{g(x)}\lim_{x\rightarrow2}\frac{f'(x)}{g'(x)}$$
$$=\lim_{x\rightarrow2}\frac{\frac{d}{dx}(x^2-7x+10)}{\frac{d}{dx}(x-2)}$$
$$=\lim_{x\rightarrow2}\frac{2x-7}{1}$$
$$=\lim_{x\rightarrow2}(2x-7)$$
$$=2\times2-7$$
$$=4-7$$
$$=-3$$

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