Limit of \( \displaystyle \frac{1 - \cos{\left(2 x + 1 \right)}}{2 x + 1} \) as \( x \to - \frac{1}{2} \)

Problem 1.457 · medium

1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{1 - \cos{\left(2 x + 1 \right)}}{2 x + 1}\right) \]
   limitSet up the limit at the point where the denominator is zero.✓ Proved
2. \[ = \lim_{x \to - \frac{1}{2}^+} \text{NaN} \]
   simplifyEvaluate the numerator and denominator at x = -1/2 to identify the indeterminate form.Reviewed
3. \[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(2 x + 1 \right)}\right)}{\frac{d}{d x} \left(2 x + 1\right)}\right) \]
   lhopitalApply L'Hopital's Rule because we have a 0/0 indeterminate form.Reviewed
4. \[ = \lim_{x \to - \frac{1}{2}^+} \sin{\left(2 x + 1 \right)} \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction by canceling the constant 2.✓ Proved
5. \[ = 0 \]
   limit simplifyEvaluate the limit by substituting the point into the simplified expression. Calculate the final value.✓ Proved