Find values of a, b and c such that:
lim (cos 4x + a cos 2x + b)/x^4 = Finite
x --> 0
You have missed a 'c' somewhere, thats why we will not be getting any answer correctly. But I will tell you the general idea...
Let L be the limit.
Observe that if $\displaystyle g(0) = 0$ and $\displaystyle \lim_{x \to 0} \frac{f(x)}{g(x)} =$ L, then $\displaystyle \lim_{x \to 0} f(x) = 0 $.
Applying it here, we get $\displaystyle 1 + a + b = 0$.
Now apply L'Hospital's rule, to get another form for L. Do the same process again.
Alternate trick is to substitute the power series for cos and choosing coefficients such that the limit exists.
You can however use power series or L'Hospitals to get the following equations:
a+b+1 = 0 and 8 + 2a = 0 and thus a = -4 and b = 3. So the limit L = 8.
To do this using power series, write $\displaystyle \cos t = 1 - \frac{t^2}{2!} + \frac{t^4}{4!} + (t^6$ ke terms$\displaystyle ....)$. Then group terms with same powers in the numerator. All terms with constant and power of x^2 must go to 0. That will give you the above two equations....