Evaluate the following limit: _ x 0 +bx ax+ - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{ax}+\text{bx}}{\text{ax}+\sin\text{bx}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{ax}+\text{bx}}{\text{ax}+\sin\text{bx}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\frac{\sin\text{ax}}{\text{x}}+\text{b}}{\text{a}+\frac{\sin\text{bx}}{\text{bx}}}$ $=\frac{\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{ax}}{\text{ax}}\times\text{a}+\text{b}}{\text{a}+\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{bx}}{\text{bx}}\times\text{b}}$ $=\frac{\text{a}+\text{b}}{\text{a}+\text{b}}$ $=1$

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