Question:
Evaluate the Given limit: \(\lim_{x\rightarrow 0}\)\(\frac{sin\,ax}{bx}\)
Evaluate the Given limit: \(\lim_{x\rightarrow 0}\)\(\frac{sin\,ax}{bx}\)
Updated On: Oct 23, 2023
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Solution and Explanation
At x = 0, the value of the given function takes the form 0/0.
\(\lim_{x\rightarrow 0}\)\(\frac{sin\,ax}{bx}\)= \(\lim_{x\rightarrow 0}\) \(\frac{sin\,ax}{bx}\) \(\times\) \(\frac{ax}{bx}\)
\(\lim_{x\rightarrow 0}\) (\(\frac{sin\,ax}{bx}\)) (\(\frac{a}{b}\))
\(\frac{a}{b}\) \(\lim_{ax\rightarrow 0}\) (\(\frac{sin\,ax}{ax}\)) [x\(\rightarrow\)0 \(\Rightarrow\) ax \(\rightarrow\) 0]
=\(\frac{a}{b}\times 1\) [lim y\(\rightarrow\)0 \(\frac{sin\,y}{y}\) = 1]
= \(\frac{a}{b}\)
\(\lim_{x\rightarrow 0}\)\(\frac{sin\,ax}{bx}\)= \(\lim_{x\rightarrow 0}\) \(\frac{sin\,ax}{bx}\) \(\times\) \(\frac{ax}{bx}\)
\(\lim_{x\rightarrow 0}\) (\(\frac{sin\,ax}{bx}\)) (\(\frac{a}{b}\))
\(\frac{a}{b}\) \(\lim_{ax\rightarrow 0}\) (\(\frac{sin\,ax}{ax}\)) [x\(\rightarrow\)0 \(\Rightarrow\) ax \(\rightarrow\) 0]
=\(\frac{a}{b}\times 1\) [lim y\(\rightarrow\)0 \(\frac{sin\,y}{y}\) = 1]
= \(\frac{a}{b}\)
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