Question

What is the limit as x approaches 0 of `(2x)/tan(3x)`?

Answer 1

`2/3`

Explanation:

The first thing you should always try when calculating limits, is just entering the `x` value into the function:

`lim_{x \to 0} (2x)/(tan(3x)) = (2*0)/(tan(3*0)) = 0/tan(0)=0/0`

Seeing that plugging in gives an indeterminate form, we need to use L'Hospital's Rule.

`2lim_{x \to 0} (1)/((3csc^2(3x))/cot^2(3x))`

Note that I moved the 2 on top of the limit outside of the limit.

We can move the 3 in the denominator of the 2. `1/csc^2(x)` is `sin^2(x)` and `(1/1)/cot^2(x)` is simply `cot^2(x)`. But `cot^2(x)` is `cos^2(x)/sin^2(x)` so the `sin^2(x)` gets cancelled leaving:

`(2/3)lim_{x \to 0} cos^2(3x)`.

Cos(0)=0 so we're left with `2/3` which is our answer.

Answer 2

`lim_(x rarr 0) (2x)/tan(3x) = 2/3`

Explanation:

The limit:

`lim_(x rarr 0) (2x)/tan(3x)`

is of an indeterminate form `0/0`, and so we can apply L'Hôpital's rule which states that for an indeterminate limit then, providing the limits exits then:

`lim_(x rarr a) f(x)/g(x) = lim_(x rarr a) (f'(x))/(g'(x))`

And so applying L'Hôpital's rule we get:

`lim_(x rarr 0) (2x)/tan(3x) = lim_(x rarr 0) (2)/(3sec^2(3x))`
`" "= 2/3lim_(x rarr 0) 1/(sec^2(3x))`
`" "= 2/3`

Answer 3

`lim_(x rarr 0) (2x)/tan(3x) = 2/3`

Explanation:

Another approach that doesn't rely on using L'Hôpital's rule

We can write the limit as:

`lim_(x rarr 0) (2x)/tan(3x) = lim_(x rarr 0) (2x)cot(3x)`
`" " = 2lim_(x rarr 0) xcot(3x)`

Let us look at the Taylor Series for `cotx`, which is as follows:

`cotx=1/x-x/3-x^3/45 - ...`

And so we can write a series expansion for the limit:

`lim_(x rarr 0) (2x)/tan(3x) = 2lim_(x rarr 0) (x){1/((3x))-((3x))/3-(3x)^3/45 - ... }`
`" "= 2lim_(x rarr 0) {1/3-x^2-(27x^4)/45 - ... }`
`" "= 2(1/3)`
`" "= 2/3`

Answer 4

`lim_(xrarr0)(2x)/tan(3x) = 2lim_(xrarr0)(x/sin(3x) * cos(3x))`

`2lim_(xrarr0)(3x)/sin(3x) * lim_(xrarr0)cos(3x)`

`= 2/3lim_(xrarr0)(3x)/sin(3x) * lim_(xrarr0)cos(3x)`

`= 2/3(1)(1) = 2/3`

Answer 5

To determine the limit graphically, se the graph below.

Explanation:

graph{2x/tan(3x) [-6.17, 6.316, -2.69, 3.547]}

Zoom in (use wheel) until you can guess that the limit is about `0.667`