Evaluate the following limit: _ x - 12 8x^3+1 2 x +1 - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow-\frac12}\frac{8\text{x}^3+1}{2{\text{x}}+1}$

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Answer

$\lim\limits_{\text{x}\rightarrow-\frac12}\frac{8\text{x}^3+1}{2{\text{x}}+1}=\lim\limits_{\text{x}\rightarrow-\frac12}\frac{8\Big(\text{x}^2+\frac18\Big)}{2\Big(\text{x}+\frac12\Big)}$ $=\frac82\lim\limits_{\text{x}\rightarrow-\frac12}\frac{\Big(\text{x}^3+\big(\frac12\big)^3\Big)}{\text{x}+\frac12}$ $=4\lim\limits_{\text{x}\rightarrow-\frac{1}{2}}\frac{\Big(\text{x}+\frac12\Big)\Big(\text{x}^2+\frac14-\frac12\text{x}\Big)}{\Big(\text{x}+\frac12\Big)}$ $=4\Big(\Big(\frac{-1}{2}\Big)+\frac14-\frac12\Big(\frac{-1}{2}\Big)\Big)$ $=4\Big(\frac14+\frac14+\frac14\Big)$ $=3$

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