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Evaluate the limit. lim_{x rightarrow 3}frac{x^{3}-27}{x^{2}-9}

Question
Evaluate the limit.
$$\displaystyle\lim_{{{x}\rightarrow{3}}}{\frac{{{x}^{{{3}}}-{27}}}{{{x}^{{{2}}}-{9}}}}$$ZSK

Answers (1)

2021-01-29
For these sorts of problems, you always need to cancel something from the numerator and denominator before the limit can be evaluated. In this case, notice that
$$\displaystyle{x}^{{{2}}}−{9}={\left({x}−{3}\right)}{\left({x}+{3}\right)}{\quad\text{and}\quad}{x}^{{{3}}}−{27}={\left({x}−{3}\right)}{\left({x}^{{{2}}}+{3}{x}+{9}\right)}$$
So the common factor that can be cancelled is x−3.
$$\displaystyle\lim_{{{x}\rightarrow{3}}}{\frac{{{x}^{{{3}}}-{27}}}{{{x}^{{{2}}}-{9}}}}=\lim_{{{x}\rightarrow{3}}}{\frac{{{x}^{{{2}}}+{3}{x}+{9}}}{{{x}+{3}}}}$$
And now at this point, we can use the fact that this function is continuous at the point x=3 to substitute in 3.
$$\displaystyle\lim_{{{x}\rightarrow{3}}}{\frac{{{x}^{{{2}}}+{3}{x}+{9}}}{{{x}+{3}}}}={\frac{{{3}^{{{2}}}+{9}+{9}}}{{{3}+{3}}}}={\frac{{{9}}}{{{2}}}}$$

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