Question:

Evaluate the Given limit: limx0\lim_{x\rightarrow 0} sinax =bxax\frac{bx}{ax}+ sinbx a,b,a+b ≠ 0

Updated On: Oct 23, 2023
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Solution and Explanation

limx0sinax\lim_{x\rightarrow 0}\,sin\,ax = bxax+sinbx\frac{bx}{ax}+sin\,bx
At x = 0, the value of the given function takes the form 0/0.
Now,
limx0\lim_{x\rightarrow 0} sinaxsin\,ax = bxax+sinbx\frac{bx}{ax}+sin\,bx
limx0\lim_{x\rightarrow 0} (sinaxax\frac{sin\,ax}{ax}) ax + bxax\frac{bx}{ax} + bx(sinbxbx\frac{sin\,bx}{bx})
=(limax0\lim_{ax\rightarrow 0} sinaxax\frac{sin\,ax}{ax}×\times limx0\lim_{x\rightarrow 0}(ax) + limx0\lim_{x\rightarrow 0}bx / limx0\lim_{x\rightarrow 0}(ax) + (limbx0\lim_{bx\rightarrow 0} sinbxbx\frac{sin\,bx}{bx}) [As x\rightarrow0\Rightarrowax\rightarrow0 and bx\rightarrow0]
=limx0(ax)limx0bxlimx0ax+limx0bx\frac{\lim_{x\rightarrow 0}(ax)\lim_{x\rightarrow 0}bx}{\lim_{x\rightarrow 0}ax+\lim_{x\rightarrow 0}bx}
=limx0(ax+bx)limx0(ax+bx)\frac{\lim_{x\rightarrow 0}(ax+bx)}{\lim_{x\rightarrow 0}(ax+bx)}
=limx0\lim_{x\rightarrow 0} (1)
=1
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