91Ó°ÊÓ

In calculus, indeterminate forms often occur when evaluating limits. An indeterminate form, such as \( \frac{0}{0} \), doesn’t provide enough information to determine the limit's value without further analysis. When you see an indeterminate form, it signals that you need a different strategy to find the limit. This might involve algebraic manipulation, L'Hôpital's rule, or factoring to simplify the expression.

In this exercise, substituting \( t = -2 \) directly into the given expression \( \frac{t^3 + 8}{t+2} \) results in the indeterminate form \( \frac{0}{0} \). Recognizing this helps you decide that additional algebraic steps, like factoring, are necessary to solve the problem and successfully evaluate the limit.

The sum of cubes is an algebraic identity used to simplify expressions of the form \( a^3 + b^3 \). The formula is \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \). This can help in breaking down the cube into a product of factors, which is quite useful when dealing with indeterminate forms like \( \frac{0}{0} \).

In our exercise, we have \( t^3 + 8 \), which can be rewritten as \( t^3 + 2^3 \). By applying the sum of cubes formula, we simplify it to \( (t+2)(t^2-2t+4) \). This process is crucial because it allows us to cancel out common factors and resolve the indeterminate form.

Factoring polynomials involves rewriting a polynomial as a product of smaller polynomials. This technique is essential in calculus, especially when evaluating limits that present indeterminate forms.

In the exercise discussed, factoring is applied to the numerator \( t^3 + 8 \). After recognizing it as a sum of cubes, it’s expressed as \( (t+2)(t^2-2t+4) \) using the specific sum of cubes formula.

• This factored form allows us to cancel the common term \( (t+2) \) in both the numerator and denominator.
• Once this cancellation is done, the expression simplifies, helping us accurately find the limit as \( t \) approaches -2 without resulting in an undefined expression.

Thus, factoring makes complex problems more manageable and paves the way for simple mathematical operations, aiding in the clear calculation of limits.