What is the correct option of the following question?
The answer is
For this problem I assume that
Let's rewrite the denominator in terms of sine and cosine exclusively.
#secx - cosx= 1/cosx - cosx = (1 -cos^2x)/cosx = sin^2x/cosx#
We can also use the laws of logarithms to simplify the numerator to
#ln[(1 + x + x^2)(1 - x + x^2)]#
Now let's rewrite the limit.
#L = lim_(x-> 0) (ln[(1 +x +x^2)(1 -x + x^2)])/(sin^2x/cosx)#
#L = lim_(x->0) (cosxln[(1 +x + x^2)(1 - x+ x^2)])/sin^2x#
If we try to evaluate the limit now, we will get
Finding the derivative of the numerator may be a little bit long. We must first differeintaite
Start by expanding within the brackets:
#1 - x + x^2 + x - x^2 + x^3 + x^2 - x^3 + x^4#
Now combine like terms
#x^4 + x^2 + 1#
Now differentiate using the chain and power rules.
#d/dx(ln(x^4 + x^2 + 1)) = (4x^3 + 2x)/(x^4 + x^2 + 1)#
Now use the product rule to differentiate the entire numerator.
#L = lim_(x->0) (-sinxln(x^4 +x^2 + 1) + cosx((4x^3 + 2x)/(x^4 + x^2 + 1)))/(sin(2x))#
You will find that this gives
It's tedious, but in the end you should get an answer of
Hopefully this helps!