Question

lim x 0 cot x/cot 2x
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Answer 1

As x approaches 0 this function approaches 0/0 which is indeterminate

So use L'Hopitals Rule

lim x>0 cot x / cot 2x lim x>0 d/dx (cotx) / d/dx (cot2x) .

and keep applying LHopitals rule until you get a function that is NOT 0/0 or ∞ /∞

ʜᴏᴘᴇ ɪᴛ ʜᴇʟᴘs ʏᴏᴜ✌

Answer 2

The solution of limₓ₋₀ (cot x/cot 2x) is 1/2.

Given:

limₓ₋₀ (cot x/cot 2x)

To Find:

We are required to find the solution for the given equation.

Solution:

f(x) = cot x, g(x) = cot 2x

limₓ₋₀ (f(x)/g(x)) = limₓ₋₀ (cot x/cot 2x) --------(1)

x tends to zero in the limit then substitute x = 0 in equation(1), we get

limₓ₋₀ (f(x)/g(x)) = cot 0/ cot 2×0

= ∞/∞

The solution was in the form of ∞/∞ known as indeterminate form.

According to L'Hospital's Rule:

"If we have an indeterminate form 0/0 or ∞/∞ all we need to do is differentiate the numerator and differentiate the denominator and then take the limit."

f '(x) = -cosec²x

g '(x) = -2cosec²x

limₓ₋₀ (cot x/cot 2x) = limₓ₋₀ (f '(x)/g '(x))

= limₓ₋₀ (-cosec²x/-2cosec²x)

= limₓ₋₀ (cosec²x/2cosec²x)

=  limₓ₋₀ (1/2)

= 1/2

Therefore, The solution of limₓ₋₀ (cot x/cot 2x) is 1/2.

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