limit (x -> 0): (cos x - 1) / x
The answer is 0. I can see this with graphing calculator, but how do I solve algebraically?
For cosine of x, change it to the infinite series, then divide by x, and you have it.
http://www.ucl.ac.uk/Mathematics/geomath/level2/series/ser11.html
To solve this algebraically, we can use the concept of limits and the trigonometric identity. Let's break it down step by step:
1. Begin by factoring out a common factor of -1 from the numerator.
(cos x - 1) / x = (-1)(1 - cos x) / x
2. Next, we can rewrite 1 - cos x using a trigonometric identity. The identity we'll use is:
1 - cos x = 2 sin^2 (x/2)
By substituting this back into the expression, we get:
(-1)(2 sin^2 (x/2)) / x
3. Now, cancel out the common factors of -1 and 2:
-2 sin^2 (x/2) / x
4. We can further simplify by factoring out sin^2 (x/2):
-2 (sin (x/2))^2 / x
Now, let's consider the limit as x approaches 0. As x approaches 0, (x/2) also approaches 0, so we can rewrite the expression as:
-2 (sin (x/2))^2 / (x/2) * 2
Notice that sin (x/2) / (x/2) approaches 1 as x approaches 0.
Therefore, the limit simplifies to:
-2 * 1^2 * 2 = -4
Hence, the limit of (cos x - 1) / x as x approaches 0 is -4.