$\lim\limits_{x\to 1}f(x)$
$=\lim\limits_{x\to 1}\dfrac{\sqrt{5x^2+3x+1}-(4x-1)}{x-1}$
$=\lim\limits_{x\to 1}\dfrac{5x^2+3x+1-16x^2+8x-1}{(x-1)(\sqrt{5x^2+3x+1}+4x-1)}$
$=\lim\limits_{x\to 1}\dfrac{-11x^2+11x}{(x-1)(\sqrt{5x^2+3x+1}+4x-1)}$
$=\lim\limits_{x\to 1}\dfrac{-11x}{\sqrt{5x^2+3x+1}+4x-1}$
$=\dfrac{-11}{\sqrt{5+3+1}+4-1}$
$=\dfrac{-11}{6}$
$f(1)=a$
$\to a=\dfrac{-11}{6}$
