Question

Evaluate the limit as x approaches 0 of (cos(x) - 1) / x² + 28x⁴ / x². (Hint: Use power series.)

Answer 1

To evaluate the limit as x approaches 0 of (cos(x) - 1) / x² + 28x⁴ / x², we can use the power series for cosine and simplify the expression.

Step-by-step explanation:

To evaluate the limit as x approaches 0 of (cos(x) - 1) / x² + 28x⁴ / x², we can use the power series for cosine, which is given by cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Substituting this into the expression, we get (1 - x²/2! + x⁴/4! - x⁶/6! + ... - 1) / x² + (28x⁴ / x²).

Simplifying, we have (-x²/2! + x⁴/4! - x⁶/6! + ...) / x² + (28x²). Using the fact that x² cancels out, we are left with -1/2! + x²/4! - x⁴/6! + ... + 28.

Taking the limit as x approaches 0, all the terms with x in the numerator disappear, leaving us with the value 28. Therefore, the limit as x approaches 0 of (cos(x) - 1) / x² + 28x⁴ / x² is equal to 28.