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

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

Question
Limits and continuity
asked 2020-11-09
Find the limits:
\(\lim_{x\rightarrow2}\frac{x^2-7x+10}{x-2}\)

Answers (1)

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.
\(\lim_{x\rightarrow2}\frac{f(x)}{g(x)}\lim_{x\rightarrow2}\frac{f'(x)}{g'(x)}\)
\(=\lim_{x\rightarrow2}\frac{\frac{d}{dx}(x^2-7x+10)}{\frac{d}{dx}(x-2)}\)
\(=\lim_{x\rightarrow2}\frac{2x-7}{1}\)
\(=\lim_{x\rightarrow2}(2x-7)\)
\(=2\times2-7\)
\(=4-7\)
\(=-3\)
0

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