Determine the following limits.


b. lim t→−2^− t^3 − 5t^2 + 6t / t^4 − 4t^2

Verified step by step guidance

1. Step 1: Identify the type of limit problem. This is a rational function limit as \( t \) approaches \(-2^-\), which means we are approaching \(-2\) from the left.
2. Step 2: Factor the numerator and the denominator if possible. The numerator is \( t^3 - 5t^2 + 6t \) and the denominator is \( t^4 - 4t^2 \).
3. Step 3: Factor out common terms. For the numerator, factor out \( t \) to get \( t(t^2 - 5t + 6) \). For the denominator, factor out \( t^2 \) to get \( t^2(t^2 - 4) \).
4. Step 4: Simplify the expression. The numerator \( t(t^2 - 5t + 6) \) can be further factored as \( t(t-2)(t-3) \). The denominator \( t^2(t^2 - 4) \) can be factored as \( t^2(t-2)(t+2) \).
5. Step 5: Cancel common factors. Cancel the common factor \( (t-2) \) from the numerator and the denominator, then evaluate the limit of the simplified expression as \( t \) approaches \(-2^-\).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In the given limit, both the numerator and denominator are polynomials. Understanding their behavior, especially as the variable approaches specific values, is essential for limit evaluation.

Factoring and Simplifying

Factoring and simplifying expressions is a key technique in calculus for resolving limits, especially when direct substitution leads to indeterminate forms like 0/0. By factoring polynomials, one can often cancel common terms, making it easier to evaluate the limit as the variable approaches a specific value.