<s>[INST] Limit using Taylor series expansion 
Can someone help me with this limit. I know I have to expand in Taylor series
$\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3+o(x^3)$
$a^x=e^{x\ln a}=1+x\ln a+\frac{1}{2!}x^2(\ln a)^2+\frac{1}{3!}x^3(\ln a)^3+o(x^3)$.
But how to proceed next?
 [/INST] Note that from Taylor's expansion
$$\frac{ a^{\sqrt{x+1}}-a^{1+\frac{x^2}2-\frac{x^2}8} }{x^3}
=\left(a^{1+\frac{x^2}2-\frac{x^2}8}\right)\frac{ a^{\frac{x^3}{16}+o(x^3)} -1 }{x^3}
=\left(a^{1+\frac{x^2}2-\frac{x^2}8}\right)\frac{ a^{\frac{x^3}{16}+o(x^3)} -1 }{\frac{x^3}{16}+o(x^3)}\frac{\frac{x^3}{16}+o(x^3)}{x^3}$$
now use standard limits.
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