Limits chapter!!
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![\lim_{x \to 0}\frac{a^{\tan x}-a^{\sin x}}{\tan x -\sin x}\\\;\\=\lim_{x \to 0}\frac {a^{\sin x}(a^{(\tan x-\sin x)}-1)}{\tan x-\sin x}\\\;\\=\lim_{x \to 0}\frac {a^{\sin x}[1+(\tan x-\sin x)\log_ea+\frac{(\tan x-\sin x)^{2}(log_ea)^{2}}{2!}.............-1]}{\tan x-\sin x}\\\;\\=\lim_{x \to 0}a^{\sin x}[\log_ea+\frac{(\tan x-\sin x)(log_ea)^{2}}{2!}.............]\\\;\\=a^{\sin 0}(\log_ea+ 0+0)\\\;\\=\log_ea](https://tex.z-dn.net/?f=%5Clim_%7Bx+%5Cto+0%7D%5Cfrac%7Ba%5E%7B%5Ctan+x%7D-a%5E%7B%5Csin+x%7D%7D%7B%5Ctan+x+-%5Csin+x%7D%5C%5C%5C%3B%5C%5C%3D%5Clim_%7Bx+%5Cto+0%7D%5Cfrac+%7Ba%5E%7B%5Csin+x%7D%28a%5E%7B%28%5Ctan+x-%5Csin+x%29%7D-1%29%7D%7B%5Ctan+x-%5Csin+x%7D%5C%5C%5C%3B%5C%5C%3D%5Clim_%7Bx+%5Cto+0%7D%5Cfrac+%7Ba%5E%7B%5Csin+x%7D%5B1%2B%28%5Ctan+x-%5Csin+x%29%5Clog_ea%2B%5Cfrac%7B%28%5Ctan+x-%5Csin+x%29%5E%7B2%7D%28log_ea%29%5E%7B2%7D%7D%7B2%21%7D.............-1%5D%7D%7B%5Ctan+x-%5Csin+x%7D%5C%5C%5C%3B%5C%5C%3D%5Clim_%7Bx+%5Cto+0%7Da%5E%7B%5Csin+x%7D%5B%5Clog_ea%2B%5Cfrac%7B%28%5Ctan+x-%5Csin+x%29%28log_ea%29%5E%7B2%7D%7D%7B2%21%7D.............%5D%5C%5C%5C%3B%5C%5C%3Da%5E%7B%5Csin+0%7D%28%5Clog_ea%2B+0%2B0%29%5C%5C%5C%3B%5C%5C%3D%5Clog_ea "\lim_{x \to 0}\frac{a^{\tan x}-a^{\sin x}}{\tan x -\sin x}\\\;\\=\lim_{x \to 0}\frac {a^{\sin x}(a^{(\tan x-\sin x)}-1)}{\tan x-\sin x}\\\;\\=\lim_{x \to 0}\frac {a^{\sin x}[1+(\tan x-\sin x)\log_ea+\frac{(\tan x-\sin x)^{2}(log_ea)^{2}}{2!}.............-1]}{\tan x-\sin x}\\\;\\=\lim_{x \to 0}a^{\sin x}[\log_ea+\frac{(\tan x-\sin x)(log_ea)^{2}}{2!}.............]\\\;\\=a^{\sin 0}(\log_ea+ 0+0)\\\;\\=\log_ea")
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Anonymous:
Bhaiya I need both the questions solution.
ok ! will add a attachment soon
Thank you Bhaiya :-)
ans. is D
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Correct answer Bhaiya. Thank you :-)
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