What analytic method steps are required to find the limit of cos x-1/2x^2 ? Limit greater than 0. Estimating -0.25 but stalled ....
To find the limit of the given function, cos(x) - 1/(2x^2), as x approaches a specific value greater than zero, you can follow these steps:
1. Identify the value that x is approaching: In this case, you mentioned x is approaching a limit greater than 0.
2. Substitute the given value into the function: Replace x with the given value in the function cos(x) - 1/(2x^2).
3. Simplify the expression: Evaluate the expression using the substituted value.
4. If the expression simplifies to a finite value, then that value is the limit. If it does not simplify or approaches infinity or negative infinity, further steps may be required.
In your case, you estimated the limit to be approximately -0.25 but were stalled. Let's continue the steps:
5. Apply the limit operations: Since the function includes trigonometric and polynomial terms, we can use basic limit properties to simplify the expression. For the first term, cos(x), we know that the limit of cosine function as x approaches any value is well-defined and finite. For the second term, 1/(2x^2), we can simplify further by applying the limit operation.
6. Take the limit of each term separately:
- The limit of cos(x) as x approaches any value is between -1 and 1. So, the limit of cos(x) - 1 will be between -2 and 0.
- The limit of 1/(2x^2) as x approaches any positive value is finite and equal to zero (since the denominator becomes infinitely large).
7. Combine the limits: The limit of the given function is the combination of the limits of each term. In this case, it will be between -2 and 0.
Therefore, based on the steps outlined above, the limit of cos(x) - 1/(2x^2) as x approaches a value greater than 0 is between -2 and 0.