lim x→0 of [2-cos3x-cos4x]/(x)
as y--->0
cos y = 1 - y^2/2 + ........
numerator:
2 - 1 +(1/2)(9)x^2 -1 +(1/2)(16)x^2
0 + (25/2)x^2 oh my
now divide by x
25 x/2
goes to 0 as x goes to 0
To find the limit of the given expression as x approaches 0, we can apply the concept of L'Hospital's Rule. L'Hospital's Rule states that if the limit of the ratio of two functions exists but is indeterminate (such as 0/0 or ∞/∞), then taking the derivative of both the numerator and denominator and then evaluating the limit will give the same result.
Let's start by evaluating the derivative of the numerator and denominator separately.
1. Derivative of the numerator:
The numerator is 2 - cos(3x) - cos(4x). Taking the derivative of each term, the first term, which is a constant, gives us zero since the derivative of a constant is zero. For the remaining terms, we will use the chain rule.
The derivative of cos(3x) is -sin(3x) multiplied by the derivative of the argument, which is 3.
So, the derivative of cos(3x) is -3sin(3x).
Similarly, the derivative of cos(4x) is -4sin(4x) since the derivative of the argument, which is 4x, is actually 4.
Therefore, the derivative of the numerator is (-3sin(3x) - 4sin(4x)).
2. Derivative of the denominator:
The derivative of x is simply 1.
Now that we have obtained the derivative of both the numerator and denominator, let's evaluate the limit by plugging in x = 0, which gives:
lim x→0 of (-3sin(3x) - 4sin(4x)) / 1
Since sin(0) = 0, we can substitute x with 0 in the numerator to simplify the expression further:
lim x→0 of (-3sin(0) - 4sin(0)) / 1
= lim x→0 of (0 - 0) / 1
= lim x→0 of 0 / 1
= 0.
Therefore, the limit of the given expression as x approaches 0 is 0.
To evaluate the limit of the expression (2 - cos(3x) - cos(4x))/(x) as x approaches 0, we can use L'Hôpital's Rule.
Step 1: Differentiate the numerator and denominator separately.
Differentiating the numerator:
The derivative of 2 is 0.
The derivative of cos(3x) is -3sin(3x) using the chain rule.
The derivative of cos(4x) is -4sin(4x) using the chain rule.
Differentiating the denominator:
The derivative of x is 1.
So, the expression becomes:
lim x→0 of [-3sin(3x) - 4sin(4x)] / 1.
Step 2: Substitute x = 0 into the expression.
Substituting x = 0 into the expression:
[-3sin(3(0)) - 4sin(4(0))] / 1
= [-3sin(0) - 4sin(0)] / 1
= [0 - 0] / 1
= 0.
Therefore, the limit of the expression (2 - cos(3x) - cos(4x))/(x) as x approaches 0 is 0.