This question can be solved primarily by L'Hospital Rule and the Product Rule.
[tex]y= \lim_{x \to 0} \frac{x^2cos(x)-sin^2(x)}{x^4} [/tex]
I) Product Rule and L'Hospital Rule:
[tex]y= \lim_{x \to 0} \frac{[2xcos(x)-x^2sin(x)]-2sin(x)cos(x)}{4x^3} [/tex]
II) Product Rule and L'Hospital Rule:
[tex]y= \lim_{x \to 0} \frac{[-2xsin(x)+2cos(x)]-[2xsin(x)+x^2cos(x)]-[2cos^2(x)-2sin^2(x)]}{12x^2} \\ y= \lim_{x \to 0} \frac{2cos(x)-4xsin(x)-x^2cos(x)-2cos^2(x)+2sin^2(x)}{12x^2}[/tex]
III) Product Rule and L'Hospital Rule:
[tex]]y= \alpha + \beta \\ \\ \alpha =\lim_{x \to 0} \frac{-2sin(x)-[4sin(x)+4xcos(x)]-[2xcos(x)-x^2sin(x)]}{24x} \\ \beta = \lim_{x \to 0} \frac{4sin(x)cos(x)+4sin(x)cos(x)}{24x} \\ \\ y = \lim_{x \to 0} \frac{-6sin(x)-4xcos(x)-2xcos(x)+x^2sin(x)+8sin(x)cos(x)}{24x}[/tex]
IV) Product Rule and L'Hospital Rule:
[tex]y = \phi + \varphi \\ \\ \phi = \lim_{x \to 0} \frac{-6cos(x)-[-4xsin(x)+4cos(x)]-[2cos(x)-2xsin(x)]}{24x} \\ \varphi = \lim_{x \to 0} \frac{[2xsin(x)+x^2cos(x)]+[8cos^2(x)-8sin(x)]}{24x} [/tex]
V) Using the Definition of Limit:
[tex]y= \frac{-6*1-4*1-2*1+8*1^2}{24} \\ y= \frac{-4}{24} \\ \boxed {y= \frac{-1}{6} }[/tex]
