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

Problem 1.376 · medium

1. \[ \lim_{x \to -1^+}\left(\frac{- x + e^{x + 1} - 2}{\left(x + 1\right)^{2}}\right) \]
   limit algebraStart with the limit of the given function. Rewrite the numerator to group terms with (x + 1).✓ Proved
2. \[ = \lim_{x \to -1^+}\left(\frac{\frac{d}{d x} \left(- x + e^{x + 1} - 2\right)}{\frac{d}{d x} \left(x + 1\right)^{2}}\right) \]
   lhopitalApply L'Hopital's rule since the limit is in the form 0/0.✓ Proved
3. \[ = \lim_{x \to -1^+}\left(\frac{e^{x + 1} - 1}{2 x + 2}\right) \]
   simplifyCompute the derivatives in the numerator and denominator.✓ Proved
4. \[ = \lim_{x \to -1^+}\left(\frac{\frac{d}{d x} \left(e^{x + 1} - 1\right)}{\frac{d}{d x} \left(2 x + 2\right)}\right) \]
   lhopitalApply L'Hopital's rule again since the limit is in the form 0/0.✓ Proved
5. \[ = \lim_{x \to -1^+}\left(\frac{e^{x + 1}}{2}\right) \]
   simplifyCompute the derivatives.✓ Proved
6. \[ = \frac{1}{2} \]
   limit simplifyEvaluate the limit by substituting x = -1. Simplify the final value.✓ Proved