<s>[INST] Indeterminate form limits question $$\lim_{x\to 0}\frac{10^x - 2^x - 5 ^ x + 1 } {x\tan x} $$
This is an indeterminate limit. 
I want help in solving this problem. Thanks in advance
 [/INST] If we replace in the expression $-1$ by $1$ we have in this case the indeterminate form $\frac 00$ and we calculate the limit using the Taylor series:
$$\frac{10^x - 2^x - 5 ^ x + 1 } {x\tan x}=\frac{\exp(x\log10) -\exp(x\log2) - \exp(x\log5) + 1 } {x\tan x}\sim_0\frac{1+x\log10+\frac{x^2\log^210}{2}-1-x\log2-\frac{x^2\log^22}{2}-1-x\log5-\frac{x^2\log^25}{2}+1}{x^2}\sim_0\frac{\log^210-\log^22-\log^25}{2}$$
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