If x→0limx4cos(2x)+acos(4x)−b is finite, then (a + b) is equal to:
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Step-by-Step Solution
Step 1: Analyze the limit form
For the limit to be finite when the denominator approaches zero (x4→0), the numerator must also approach zero. This is a necessary condition for L'Hopital's Rule to be applicable multiple times.
Step 2: Apply condition for numerator to be zero
Substitute x=0 into the numerator and set it equal to zero. Since cos(0)=1, we get an equation relating a and b. This gives us the first condition for the limit to be finite.
Step 3: Use Maclaurin series expansion
To evaluate the limit, we can use the Maclaurin series expansion for cos(kx). We need terms up to x4 because the denominator is x4. The general expansion for cos(u) is 1−2u2!+4u4!−….
Step 4: Substitute series into the expression
Substitute the Maclaurin series for cos(2x) and cos(4x) into the numerator. Also, substitute b=1+a from our first condition. This allows us to simplify the expression and cancel out constant terms.
Step 5: Simplify the expression
After substituting b=1+a, the constant terms 1+a-b cancel out. We then group terms by powers of x. For the limit to be finite, the coefficients of x2 in the numerator must also be zero, otherwise, the limit would be infinite.
Step 6: Set coefficient of x2 to zero
For the limit to be finite, the coefficient of x2 in the numerator must be zero. If it were not zero, the expression would simplify to a term like C/x2, which would tend to infinity as x→0. Setting the coefficient to zero allows us to solve for a.
Step 7: Calculate b and a+b
Now that we have the value of a, we can find b using the relation b=1+a. Finally, we calculate the sum a+b.