solution for limit as delta x approaches 0 (1-cos4x/1-cos2x)
however, a more careful reading reveals :-)
lim (1-cos4x)/(1-cos2x)
= lim 4sin4x/2sin2x
= lim (4sin2x cos2x)/sin2x
= lim 4cos2x
= 4
Ah, the limit as delta x approaches 0, always a party! Now, let's have some fun with this one.
Firstly, let's rewrite the expression:
(1 - cos(4x))/(1 - cos(2x))
Now, we want to simplify this expression as delta x approaches 0. When delta x is really close to 0, x is also really close to 0.
So, let's evaluate the limits individually:
lim(x->0) (1 - cos(4x)) = 0
lim(x->0) (1 - cos(2x)) = 0
Both the numerator and denominator approach 0 as delta x approaches 0. So, we can apply L'Hopital's rule to find the limit of the ratio:
lim(x->0) (1 - cos(4x))/(1 - cos(2x)) = lim(x->0) (4sin(4x))/(2sin(2x))
Now, let's plug in x = 0:
lim(x->0) (4sin(4x))/(2sin(2x)) = (4sin(0))/(2sin(0))
Wait a minute, sin(0) is 0, which means the denominator will be 0. And you know what happens when you divide by zero, right? Chaos ensues!
So, unfortunately, this limit is undefined. That's right, the party got canceled!
But don't worry, there are plenty of other limits you can calculate. Just let me know if you want to explore another one!
To find the limit as delta x approaches 0 of (1 - cos(4x))/(1 - cos(2x)), we can use a trigonometric identity to simplify the expression. The identity we'll use is:
cos(2x) = 1 - 2sin^2(x)
Using this identity, let's simplify the expression:
(1 - cos(4x))/(1 - cos(2x))
= (1 - cos(4x))/(1 - (1 - 2sin^2(x)))
= (1 - cos(4x))/(2sin^2(x))
Next, we'll simplify further:
Using another trigonometric identity, sin^2(x) = (1 - cos(2x))/2:
= (1 - cos(4x))/2((1 - cos(2x))/2)
= (1 - cos(4x))/(1/2)(1 - cos(2x))
= 2(1 - cos(4x))/(1 - cos(2x))
Now, we can take the limit as delta x approaches 0. Since this is an algebraic expression, we can substitute 0 for delta x directly:
= 2(1 - cos(4*0))/(1 - cos(2*0))
= 2(1 - cos(0))/(1 - cos(0))
= 2(1 - 1)/(1 - 1)
= 2(0)/(0)
= 0
Therefore, the limit as delta x approaches 0 of (1 - cos(4x))/(1 - cos(2x)) is 0.
To find the solution for the limit as delta x approaches 0 of the expression (1 - cos(4x))/(1 - cos(2x)), we can directly substitute 0 for delta x into the expression. However, this might result in an indeterminate form (such as 0/0) which does not provide a clear answer.
Alternatively, we can use algebraic manipulation and trigonometric identities to simplify the expression before taking the limit. Here's how we can do it:
Step 1: Apply the double-angle formula for cosine:
cos(2x) = 2cos^2(x) - 1
Step 2: Apply the double-angle formula again to the numerator:
1 - cos(4x) = 1 - cos^2(2x) = sin^2(2x)
Step 3: Substitute the rewritten expressions back into the limit:
Limit as delta x approaches 0 of (sin^2(2x))/(1 - 2cos^2(x) + 1)
Step 4: Further simplify the expression by factoring out a common term:
Limit as delta x approaches 0 of (sin^2(2x))/(2 - 2cos^2(x))
Step 5: Apply another trigonometric identity:
sin^2(2x) = 1 - cos^2(2x)
Step 6: Substitute the identity back into the limit:
Limit as delta x approaches 0 of (1 - cos^2(2x))/(2 - 2cos^2(x))
Step 7: Factor out a common term:
Limit as delta x approaches 0 of (1 - cos^2(2x))/(2(1 - cos^2(x)))
Step 8: Apply another trigonometric identity:
cos^2(2x) = 1 - sin^2(2x)
Step 9: Substitute the identity back into the limit:
Limit as delta x approaches 0 of (1 - (1 - sin^2(2x)))/(2(1 - cos^2(x)))
Step 10: Simplify the expression in the numerator:
Limit as delta x approaches 0 of sin^2(2x)/(2(1 - cos^2(x)))
At this point, we have simplified the expression as much as possible without knowing the exact values of x. To get the final solution, you would need additional context or specific values for x.
Usually the first step I take in doing a limit question is to actually substitute the approach value, so I got
lim cos 4x/(1- cos 2x) , as x -->0
--> 1/(1-0)
which is undefined,
so there is not limit, or the result is infinity
check: I tried x = .0001
and got 52631574.74