Question

Limx->0 cos5x-cos11x/cos3x-cos7x

Answer 1

we have to find $\lim_{x\to 0}\frac{cos5x-cos11x}{cos3x-cos7x}$

solution : there are many methods to solve this type of limit function. here form of limit is 0/0 so you can apply L - Hospital rule.

just differentiate numerator and denominator separately.

let's find it.

$\lim_{x\to 0}\frac{-5sin5x-(-11)sin11x}{-3sin3x-(-7)sin7x}$

⇒$\lim_{x\to 0}\frac{11sin11x-5sin5x}{7sin7x-3sin3x}$

again we get 0/0 form of limit.

so, again we can differentiate numerator and denominator separately.

$\lim_{x\to 0}\frac{121cos11x-25cos5x}{49cos7x-9cos3x}$

now put the value x = 0,

we get, the value of limit = (121 - 25)/(49 - 9)

= 96/40 = 2.4

Therefore the the value of $\lim_{x\to 0}\frac{cos5x-cos11x}{cos3x-cos7x}$ is 2.4

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