Limits And Derivatives

We are given the following limit: Step 1: Simplifying the denominator at First, factor the denominator . Start by trying as a root: Thus, is a root. We can divide the cubic polynomial by using synthetic division: Thus, the denominator can be factored as: Step 2: Substituting in the given limit Now, we can substitute the denominator into the given expression: Since is a root of the denominator, the numerator must also be zero when . Therefore, substitute into the numerator: Step 3: Substituting in the second limit Now that we know , we need to find: Substitute into the numerator and denominator: For the numerator: For the denominator: Thus, both the numerator and denominator are zero, so we apply L'Hopital's Rule. First, take the derivatives of the numerator and denominator: Numerator's derivative: Denominator's derivative: Now, substitute into these derivatives: Thus, the limit is: Thus, the value of the limit is . \bigskip