Determine the following limits.


b. $\underset{x\to 2}{\lim }\frac{x-2}{x^5-4x^3}$

Verified step by step guidance

1. Step 1: Identify the limit expression: \( \lim_{x \to 2} \frac{x-2}{x^5 - 4x^3} \).
2. Step 2: Substitute \( x = 2 \) into the expression to check if it results in an indeterminate form. Calculate \( \frac{2-2}{2^5 - 4 \cdot 2^3} = \frac{0}{0} \), which is an indeterminate form.
3. Step 3: Factor the denominator \( x^5 - 4x^3 \). Notice that \( x^3 \) is a common factor: \( x^5 - 4x^3 = x^3(x^2 - 4) \).
4. Step 4: Further factor \( x^2 - 4 \) using the difference of squares: \( x^2 - 4 = (x-2)(x+2) \). Thus, the denominator becomes \( x^3(x-2)(x+2) \).
5. Step 5: Cancel the common factor \( x-2 \) from the numerator and the denominator: \( \frac{x-2}{x^3(x-2)(x+2)} = \frac{1}{x^3(x+2)} \). Now, substitute \( x = 2 \) into the simplified expression to find the limit.

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 how functions behave near points of interest, including points where they may not be defined. For example, in the limit expression given, we are interested in how the function behaves as x approaches 2.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This is particularly useful in limit problems where direct substitution results in indeterminate forms, such as 0/0. In the given limit, factoring the denominator can help simplify the expression before evaluating the limit.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to ambiguous results, such as 0/0 or ∞/∞. These forms require further analysis or manipulation, such as factoring, to resolve. In the limit provided, substituting x = 2 directly results in an indeterminate form, necessitating additional steps to find the limit's value.