Determine the limit of the trigonometric function (if it exists).

1. lim sin x / 5x
(x -> 0)

2. lim tan^2x / x
(x ->0)

3. lim cos x tan x / x
(x -> 0)

I will be happy to critique your thinking on these.

To determine the limits of these trigonometric functions, we can use algebraic manipulation and trigonometric identities. Let's solve each one step by step.

1. lim (sin x / 5x) as x approaches 0:

To find this limit, we can apply L'Hopital's Rule. L'Hopital's Rule states that if the limit of the ratio of two functions f(x)/g(x) as x approaches a results in an indeterminate form (0/0 or ∞/∞), then the limit of the derivative of f(x) divided by the derivative of g(x) as x approaches a will be the same.

First, let's find the derivative of sin(x) and 5x:
f(x) = sin(x), f'(x) = cos(x)
g(x) = 5x, g'(x) = 5

Now we can apply L'Hopital's Rule:
lim (sin x / 5x) = lim (cos x / 5) as x approaches 0

Since cos(x) is a continuous function, we can substitute x = 0 into the expression:
lim (cos x / 5) = cos(0)/5 = 1/5

Therefore, the limit of sin x / 5x as x approaches 0 is 1/5.

2. lim (tan^2x / x) as x approaches 0:

To find this limit, we can manipulate the expression using trigonometric identities. By using the identity tan(x) = sin(x) / cos(x), we can rewrite the expression as:

lim ((sin(x) / cos(x))^2 / x) as x approaches 0

Simplifying further, we get:

lim (sin^2(x) / cos^2(x) / x) as x approaches 0

Now, let's apply L'Hopital's Rule. The derivatives are as follows:
f(x) = sin^2(x), f'(x) = 2sin(x)cos(x)
g(x) = cos^2(x), g'(x) = -2sin(x)cos(x)

Applying L'Hopital's Rule:
lim (2sin(x)cos(x)/(-2sin(x)cos(x))/x) as x approaches 0

Simplifying the expression further, we get:
lim (-1/x) as x approaches 0

As x approaches 0, -1/x approaches -∞. Therefore, the limit of tan^2x / x as x approaches 0 does not exist.

3. lim (cos(x)tan(x) / x) as x approaches 0:

To find this limit, we can manipulate the trigonometric expression using L'Hopital's Rule.

By using the identity tan(x) = sin(x) / cos(x), we can rewrite the expression as:

lim (cos(x) * (sin(x) / cos(x)) / x) as x approaches 0

Simplifying further, we get:

lim (sin(x) / x) as x approaches 0

This new limit is a well-known limit and can be evaluated directly without needing L'Hopital's Rule. The limit of sin(x)/x as x approaches 0 is equal to 1.

Therefore, the limit of cos(x)tan(x) / x as x approaches 0 is 1.