Limit of a trigonometric thing

• Jan Hill
In summary, the conversation discusses finding the limit as x approaches zero of the expression xcsc4x divided by cos17x. The suggested approach is to use trig identities and limits, such as csc(x) = 1/sin(x) and lim sin(x)/x = 0, as x --> 0. The solution involves multiplying both the numerator and denominator by 4 and using the limit lim as x approaches 0 of x/sin(4x) to deal with the denominator.
Jan Hill

Homework Statement

Find the limit as x approaches zero of xcsc4x = the numerator over cos17x = the denominator

The Attempt at a Solution

I don't know how to start except maybe multiplying both numerator and denominator by 4

This?
$$\lim_{x \to 0} \frac{x~csc(4x)}{cos(17x)}$$

Start with some trig identities, such as csc(x) = 1/sin(x). There are some trig limits that will be helpful, such as lim sin(x)/x = 0, as x --> 0.

so does this become lim as x approaches 0 of x times 1/sin(4x)
and this is lim as x approaches 0 of x/sin(4x)

and what trig limit should I use to deal with the denominator?

and please accept my many thanks for your quick reply of earlier!

Jan Hill said:
so does this become lim as x approaches 0 of x times 1/sin(4x)
and this is lim as x approaches 0 of x/sin(4x)
That's part of it.

It would be helpful if you wrote actual mathematics equations and expressions, rather than verbose English descriptions.
Jan Hill said:
and what trig limit should I use to deal with the denominator?

and please accept my many thanks for your quick reply of earlier!

What is the limit of a trigonometric function?

The limit of a trigonometric function is the value that the function approaches as the input values get closer and closer to a particular value. It is also known as the "end behavior" of the function.

How do you find the limit of a trigonometric function?

To find the limit of a trigonometric function, you can use algebraic manipulation, graphing, or substitution. The method used depends on the complexity of the function and the tools available.

What are the common trigonometric functions?

The common trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. These functions are used to relate the angles and sides of a right triangle.

What is the difference between a left and right-hand limit of a trigonometric function?

The left-hand limit of a trigonometric function is the value that the function approaches as the input values get closer and closer to a particular value from the left side. The right-hand limit is the value that the function approaches from the right side.

How are trigonometric limits used in real-world applications?

Trigonometric limits are used in various fields such as physics, engineering, and architecture to model and solve real-world problems. For example, they can be used to calculate the maximum height of a projectile or the length of a bridge's support cables.

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