Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{-\frac{1}{2}}}\frac{8\text{x}^{3}+1}{2\text{x}+1}$
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Answer
$\lim\limits_{\text{x}\rightarrow{-\frac{1}{2}}}\frac{8\text{x}^{3}+1}{2\text{x}+1}$$=\frac82\lim\limits_{\text{x}\rightarrow{-\frac{1}{2}}}\frac{\text{x}^{3}+\big(\frac12\big)^3}{\text{x}+\frac12}$
$=4\lim\limits_{\text{x}\rightarrow{-\frac{1}{2}}}\frac{\text{x}^{3}+\big(\frac12\big)^3}{\text{x}-\big(-\frac12\big)}$
Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$
Here, n = 3, $\text{a}=\frac{-1}{2}$
$=4\lim\limits_{\text{x}\rightarrow{-\frac{1}{2}}}\frac{\text{x}^{3}+\big(\frac12\big)^3}{\text{x}-\big(-\frac12\big)}=4\times3\Big(-\frac{1}{2}\Big)^{3-1}$
$=4\times3\times\frac14$
$=3$
$=4\lim\limits_{\text{x}\rightarrow{-\frac{1}{2}}}\frac{\text{x}^{3}+\big(\frac12\big)^3}{\text{x}-\big(-\frac12\big)}$
Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$
Here, n = 3, $\text{a}=\frac{-1}{2}$
$=4\lim\limits_{\text{x}\rightarrow{-\frac{1}{2}}}\frac{\text{x}^{3}+\big(\frac12\big)^3}{\text{x}-\big(-\frac12\big)}=4\times3\Big(-\frac{1}{2}\Big)^{3-1}$
$=4\times3\times\frac14$
$=3$
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