# Finding a Trig Limit by hand, no L'Hopitals

• gorlitsa
In summary, the problem involves finding the limit as x approaches 0 of the expression (cos^2(x)-1)/(2xsin(x)). The relevant equations to use are the limit of (1-cos(x))/x as x approaches 0, which equals 0, and the limit of sin(x)/x as x approaches 0, which equals 1. By substituting 1 = cos^2(x) + sin^2(x), we can simplify the expression to (1-cos(x))/x, which allows us to use one of the known limits to solve for the answer of -.5. Trig knowledge is important in solving this problem.
gorlitsa

## Homework Statement

$\stackrel{lim}{x\rightarrow 0}\frac{cos^2x-1}{2xsinx}$

## Homework Equations

$\stackrel{lim}{x\rightarrow 0}\frac{1-cosx}{x}=0$

$\stackrel{lim}{x\rightarrow 0}\frac{sinx}{x}=1$

## The Attempt at a Solution

I found this problem online (and can't remember where). It was in a limits section, so it can be solved without using l'Hopitals rule. I gave it to my class today, and then we all got stuck. We know the answer is -.5 but can't get it algebraically. Any advice?

Remember that 1 = cos^2(x) + sin^2(x).

Now substitute and simplify and you can use one of your known limits.

Thank you! I got thrown into this class half way through the year, and while I have re-learned the calc, my trig is horrible. Thank you!

## What is a trigonometric limit?

A trigonometric limit is a mathematical concept that describes the behavior of a trigonometric function as its input values approach a specific value. It is also known as a limit of a function.

## Why is it important to find trigonometric limits by hand?

Finding trigonometric limits by hand is important because it helps to develop a deeper understanding of the underlying concepts and principles behind trigonometric functions. It also allows for more accurate calculations and a better grasp of the problem at hand.

## What is the process for finding a trigonometric limit by hand?

The process for finding a trigonometric limit by hand involves using various algebraic and trigonometric identities to simplify the function and then evaluating the limit by substituting the specific value into the simplified function.

## What are some common techniques used to evaluate trigonometric limits?

Some common techniques used to evaluate trigonometric limits include factoring, rationalizing, and using trigonometric identities such as the Pythagorean identity and double angle formulas.

## What are the limitations of using L'Hopital's rule to find trigonometric limits?

L'Hopital's rule has limitations when it comes to finding trigonometric limits because it can only be applied to indeterminate forms such as 0/0 or infinity/infinity. It also does not work for all trigonometric functions and may not always give the most accurate result.

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