Evaluate the following limits in Exercise: _ x 0 +bx ax+ a,b,a+b …

Question

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{ax}+\text{bx}}{\text{ax}+\sin\text{bx}}\text{a},\text{b},\text{a}+\text{b}\neq0,$

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Answer

At x = 0, the value of the given function takes the form $\frac{0}{0}.$ Now, $\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{ax}+\text{bx}}{\text{ax}+\sin\text{bx}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\bigg(\frac{\sin\text{ax}}{\text{ax}}\bigg)\text{ax}+\text{bx}}{\text{ax}+\text{bx}\bigg(\frac{\sin\text{bx}}{\text{bx}}\bigg)}$ $ =\frac{\Bigg(\lim\limits_{\text{ax}\rightarrow0}\frac{\sin\text{ax}}{\text{ax}}\Bigg)\times\lim\limits_{\text{x}\rightarrow0}(\text{ax})+\lim\limits_{\text{x}\rightarrow0}\text{bx}}{\lim\limits_{\text{x}\rightarrow0}\text{ax}+{\lim\limits_{\text{x}\rightarrow0}\text{bx}\Bigg({\lim\limits_{\text{bx}\rightarrow0}\frac{\sin\text{bx}}{\text{bx}}}\Bigg)}} $ $[\text{As x}\rightarrow0\Rightarrow\text{ax}\rightarrow\text{and}\text{bx}\rightarrow0]$ $=\frac{\lim\limits_{\text{x}\rightarrow0}(\text{ax})+\lim\limits_{\text{x}\rightarrow0}\text{bx}}{\lim\limits_{\text{x}\rightarrow0}(\text{ax})+\lim\limits_{\text{x}\rightarrow0}(\text{bx})} \bigg[\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\bigg]$ $=\frac{\lim\limits_{\text{x}\rightarrow0}(\text{ax}+\text{bx})}{\lim\limits_{\text{x}\rightarrow0}(\text{ax}+\text{bx})}$ $= \lim\limits_{\text{x}\rightarrow0}(1)$ $=1$