Determine the following limits.


$\underset{z\to 4}{\lim }\frac{z-5}{{(z^2-10z+24)}^2}$

Verified step by step guidance

1. Step 1: Identify the limit expression: \( \lim_{{z \to 4}} \frac{{z-5}}{{(z^2 - 10z + 24)^2}} \).
2. Step 2: Factor the quadratic expression in the denominator: \( z^2 - 10z + 24 \). This can be factored as \((z-4)(z-6)\).
3. Step 3: Substitute the factored form into the limit expression: \( \lim_{{z \to 4}} \frac{{z-5}}{{((z-4)(z-6))^2}} \).
4. Step 4: Notice that direct substitution of \( z = 4 \) results in an indeterminate form \( \frac{0}{0} \). Apply L'Hôpital's Rule or simplify the expression by canceling common factors.
5. Step 5: Simplify the expression by canceling the common factor \( z-4 \) from the numerator and the denominator, then re-evaluate 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. In this case, we are interested in the limit of a fraction as the variable z approaches 4.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. This is crucial for simplifying expressions, especially when evaluating limits, as it can help eliminate indeterminate forms like 0/0. In the given limit, factoring the denominator will be necessary to simplify the expression.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not have a clear value, such as 0/0 or ∞/∞. These forms require further analysis, often through algebraic manipulation or L'Hôpital's rule, to resolve. In this limit problem, substituting z = 4 directly results in an indeterminate form, necessitating further steps to find the limit.