The limit as x approaches 0 of
(tan2x)/3x = ?
we know that
lim tanØ/Ø , as Ø ---> 0 = 1
so change your question to resemble that expression
let Ø = 2x
x = Ø/2
3x = 3Ø/2
then tan 2x/(3x) = tanØ/(3Ø/2
= (2/3) ( tanØ/Ø)
so lim 2x/(3x) = (2/3)(1) = 2/3
To find the limit as x approaches 0 of (tan2x)/(3x), we can use some trigonometric identities and algebraic manipulations.
Let's start by simplifying the expression. Using the double angle formula for tangent, we have:
tan2x = 2tan(x) / (1 - tan^2(x))
Substituting this into the original expression gives:
(2tan(x) / (1 - tan^2(x))) / (3x)
Next, we can simplify further by multiplying the expression by its conjugate, (1 - tan^2(x))/(1 - tan^2(x)):
(2tan(x) / (1 - tan^2(x))) * ((1 - tan^2(x)) / (3x))
This simplifies to:
(2tan(x) * (1 - tan^2(x))) / (3x * (1 - tan^2(x)))
Now, notice that (1 - tan^2(x)) can be written as sec^2(x) (from the trigonometric Pythagorean identity). We can substitute sec^2(x) into the expression:
(2tan(x) * sec^2(x)) / (3x * sec^2(x))
The sec^2(x) terms cancel out, leaving us with:
(2tan(x)) / (3x)
Now, we can take the limit as x approaches 0. Plugging in 0 for x gives an indeterminate form of 0/0. To evaluate this limit, we can use L'Hôpital's rule, which states that if the limit of the ratio of two functions is of the form 0/0 or ∞/∞, then the limit of their derivatives is the same as the limit of the original ratio.
So, let's find the derivative of the numerator and denominator with respect to x:
Numerator:
d/dx (2tan(x)) = 2sec^2(x)
Denominator:
d/dx (3x) = 3
Now, taking the limit as x approaches 0 of the derivatives:
lim(x->0) (2sec^2(x))/3 = 2/3
Therefore, the limit as x approaches 0 of (tan2x)/(3x) is 2/3.
To find the limit as x approaches 0 of (tan(2x))/(3x), we can use some trigonometric identities and algebraic manipulation.
Let's commence:
Step 1: Rewrite tan(2x) using the double angle formula.
tan(2x) = 2tan(x) / (1 - tan^2(x))
Step 2: Substitute the rewritten form of tan(2x) back into the original equation.
(tan(2x))/(3x) = (2tan(x) / (1 - tan^2(x)))/(3x)
Step 3: Simplify by canceling common factors.
(tan(2x))/(3x) = (2tan(x) / (1 - tan^2(x)))/(3x) = (2 / (1 - tan^2(x)))*tan(x) / (3x)
Step 4: Write tan^2(x) as (sin(x)^2 / cos(x)^2) and simplify.
(tan(2x))/(3x) = (2 / (1 - ((sin(x)^2) / (cos(x)^2))))*tan(x) / (3x)
= (2*(cos(x)^2) / (cos(x)^2 - sin(x)^2))*tan(x) / (3x)
Step 5: Use the trigonometric identity cos^2(x) - sin^2(x) = cos(2x) to simplify further.
(tan(2x))/(3x) = (2 / (cos(2x)))*tan(x) / (3x)
Step 6: Take the limit as x approaches 0.
lim(x->0) (tan(2x))/(3x) = lim(x->0) (2 / (cos(2x)))*tan(x) / (3x)
Considering the limit of each term separately:
lim(x->0) (2 / (cos(2x))) = 2 / cos(0) = 2 / 1 = 2
lim(x->0) tan(x) / (3x) = 1/3, as tan(0) = 0
Step 7: Multiply the limits calculated in the previous step.
lim(x->0) (tan(2x))/(3x) = (2)*(1/3) = 2/3
Therefore, the limit as x approaches 0 of (tan(2x))/(3x) is 2/3.