Question

Evaluate the limit of cosine 'x' divided by 'x' as 'x' approaches 0 using L'Hopital's Rule.

Answer 1

To find the limit of cos(x)/x as x approaches 0 using L'Hopital's Rule, we first recognize the indeterminate form 0/0, then differentiate both numerator and denominator to find the limit, resulting in 0.

Step-by-step explanation:

The question requires the evaluation of the limit of cos(x)/x as x approaches 0 using L'Hopital's Rule. Here's how we can proceed using the rule:

1. First, we identify that the limit x approaches 0 of cos(x)/x is an indeterminate form 0/0.
2. We then apply L'Hopital's Rule, which states that if the limit of f(x)/g(x) as x approaches a gives an indeterminate form, then the limit can be found by taking the derivative of the numerator and denominator separately.
3. Calculating the derivatives, we find the derivative of cos(x) is -sin(x) and the derivative of x is 1.
4. So, we then evaluate the limit of -sin(x)/1 as x approaches 0, which is 0.
5. Therefore, the original limit of cos(x)/x as x approaches 0 is 0.

To evaluate the limit of cosine 'x' divided by 'x' as 'x' approaches 0 using L'Hopital's Rule, we differentiate the numerator and the denominator separately. The derivative of cosine 'x' is negative sine 'x', and the derivative of 'x' is 1. Therefore, the limit becomes the limit of negative sine 'x' divided by 1 as 'x' approaches 0.

The limit of negative sine 'x' divided by 1 as 'x' approaches 0 is equal to -sin 0, which is equal to 0. Hence, the limit of cosine 'x' divided by 'x' as 'x' approaches 0 using L'Hopital's Rule is 0.