What is the limit: lim (x -> 0) (cos(x) - 1)/x^2? (2019)

What is the limit: lim (x -> 0) (cos(x) - 1)/x^2? (2019)

Step 1: Understand the limit we want to find: lim (x -> 0) (cos(x) - 1)/x^2.
Step 2: Recall the Taylor series expansion for cos(x) around x = 0: cos(x) = 1 - x^2/2 + x^4/24 - ...
Step 3: Substitute the Taylor series into the limit: cos(x) - 1 = -x^2/2 + higher order terms.
Step 4: Rewrite the limit using this substitution: lim (x -> 0) (cos(x) - 1)/x^2 = lim (x -> 0) (-x^2/2 + higher order terms)/x^2.
Step 5: Simplify the expression: lim (x -> 0) (-1/2 + higher order terms/x^2).
Step 6: As x approaches 0, the higher order terms/x^2 approach 0, so we are left with lim (x -> 0) (-1/2) = -1/2.
Step 7: Conclude that the limit is -1/2.

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