Hi everyone this is what the problem asks:
Evaluate the Limit and Justify each step by indicating the appropriate Limit Law(s).
Lim
X→0 (cos4x)/( 5+2x3 )
Is this what you're asking?
$\displaystyle \lim_{x \to 0}\frac{\cos{4x}}{5 + 2x^3}$.
If so, first note that the numerator and denominator are both continuous at $\displaystyle x = 0$, and the denominator does not $\displaystyle \to 0$.
These are the limit laws you will use...
1. If $\displaystyle f(x)$ is continuous at $\displaystyle x = a$, then $\displaystyle \lim_{x \to a}f(x) = f(a)$.
2. $\displaystyle \lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}$ if $\displaystyle g(x)$ does not $\displaystyle \to 0$.
Thus
$\displaystyle \lim_{x \to 0}\frac{\cos{4x}}{5 + 2x^3} = \frac{\lim_{x \to 0}\cos{4x}}{\lim_{x \to 0}5 + 2x^3}$
$\displaystyle = \frac{\cos{4\cdot 0}}{5 + 2(0)^3}$
$\displaystyle = \frac{1}{5}$.