How Do You Solve lim(x->0) tan(3x)/sin(8x) Using Trigonometric Limits?

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In summary, a limit is a mathematical concept used to describe the behavior of a function as the input values approach a specific value or point. To solve a limit, you can substitute the given value into the function and use various mathematical techniques. There are one-sided and two-sided limits, which consider the behavior of the function from one or both directions. To solve a trigonometric limit, you can use trigonometric identities and other techniques, such as graphing. For the specific limit "lim(x->0) tan(3x)/sin(8x)", we can use the limit laws and the fact that the limit of sin(x)/x is equal to 1 to simplify and evaluate the limit.
  • #1
donjt81
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I'm trying to solve this problem

lim(x->0) tan(3x)/sin(8x)

using the formula lim(x->0) sinx/x = 1

so i did the following

tan(3x)/sin(8x)
{sin(3x)/cos(3x)}/sin8x
{1/cos(3x)} * {sin(3x)/sin(8x)}

now I know the term {1/cos(3x)} becomes 1 when you apply the limit but I have no idea how to solve the second term which is {sin(3x)/sin(8x)}...

can anyone please help

Thanks
 
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  • #2
Hint: Multiply and divide by 24x. ;)
 
  • #3
I'd rather expand the fraction with the factor 24x.
 
  • #4
got it... thanks guys
 

FAQ: How Do You Solve lim(x->0) tan(3x)/sin(8x) Using Trigonometric Limits?

What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as the input values approach a specific value or point. It is denoted by the notation "lim" and is used to analyze the behavior of a function at a specific point or as the input values get closer and closer to that point.

How do you solve a limit?

To solve a limit, you first need to substitute the given value or variable into the function. Then, you can use algebraic manipulations, trigonometric identities, or other mathematical techniques to simplify the function and evaluate the limit. In some cases, you may need to use L'Hopital's rule or graphing to determine the limit.

What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the behavior of the function as the input values approach the specific point from one direction, either the left or the right. A two-sided limit takes into account the behavior of the function as the input values approach the point from both directions, and the limit only exists if the one-sided limits are equal.

How do you solve a trigonometric limit?

To solve a trigonometric limit, you can use trigonometric identities, such as the double-angle, half-angle, or sum and difference identities, to simplify the function. You may also need to use algebraic manipulations or L'Hopital's rule to evaluate the limit. Additionally, it can be helpful to graph the function to visualize its behavior near the limit.

How do you solve "lim(x->0) tan(3x)/sin(8x)"?

To solve this limit, we can use the fact that the limit of sin(x)/x as x approaches 0 is equal to 1. This means that we can rewrite the given function as (tan(3x)/3x) * (3x/sin(8x)). Then, we can use the limit laws to evaluate each part individually. The first part becomes 1, and the second part becomes 3/8. Therefore, the overall limit is equal to 1 * 3/8 = 3/8.

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