What is the limit of 1-2x^2 -2cosx + cos^2x all over x^2? please answer ASAP. TY
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The limit as x approaches what? Goodness.
There are different ways to approach this:
1. Since both numerator and denominator evaluate to zero as x->0, l'Hôpital's rule applies.
Differentiate the top with respect to x to get:
-2cos(x)sin(x)+2sin(x)-4x
and the bottom to get
2x
As x->0, both numerator and denominator still -> 0, thus we can apply again the rule, and differentiate:
numerator: 2*sin(x)^2-2*cos(x)^2+2*cos(x)-4
denominator: 2
As x->0, the sin(x) term vanishes, and the cos(x) terms cancel out, resulting in -4 over 2 in the denominator.
So the limit is -2.
2. If you have done series expansions before, expand numerator into a power series, taking only terms up to x^4:
1-2x^2-2(1-x^2/2+x^4/4!)+(1-x^2/2+x^4/4!)^2
=(x^8-24*x^6+144*x^4-1152*x^2)/576
Dividing by the denominator leaves us with
(x^6-24*x^4+144*x^2-1152)/576
and as x->0,
-1152/576 = -2 as before.
To find the limit of the given expression as x approaches a certain value (let's assume it is "a" in this case), we can follow these steps:
Step 1: Simplify the expression:
Start by simplifying the numerator. Apply the basic algebraic rules and trigonometric identities to simplify 1-2x^2 -2cosx + cos^2x:
1 - 2x^2 - 2cos(x) + cos^2(x) = (1 - cos^2(x)) - 2x^2 - 2cos(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite (1 - cos^2(x)) as sin^2(x):
sin^2(x) - 2x^2 - 2cos(x)
Step 2: Factor or simplify further if possible:
Before applying the limit, check if there are any possible further factorizations or simplifications to be done. In this case, the expression cannot be further simplified.
Step 3: Apply the limit:
Now we can proceed to find the limit as x approaches a:
lim(x→a) [(sin^2(x) - 2x^2 - 2cos(x))/x^2]
By substituting the value of "a" into the expression, the limit can be evaluated.
It is important to mention that without knowing the specific value of "a," we cannot provide an exact numerical answer to the limit.