Evaluate each limit and justify your answer.
lim t→4 t−4 /√t−2

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Identify the form of the limit as $ t \to 4 $. Notice that both the numerator $ t - 4 $ and the denominator $ \sqrt{t} - 2 $ approach 0, indicating an indeterminate form $ \frac{0}{0} $.

To resolve the indeterminate form, consider rationalizing the denominator. Multiply the numerator and the denominator by the conjugate of the denominator, $ \sqrt{t} + 2 $.

This gives: $ \frac{(t - 4)(\sqrt{t} + 2)}{(\sqrt{t} - 2)(\sqrt{t} + 2)} $. Simplify the denominator using the difference of squares formula: $ (\sqrt{t})^2 - 2^2 = t - 4 $.

The expression simplifies to $ \frac{(t - 4)(\sqrt{t} + 2)}{t - 4} $. Cancel the common factor $ t - 4 $ from the numerator and the denominator.

After canceling, evaluate the limit of the simplified expression $ \sqrt{t} + 2 $ as $ t \to 4 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. Evaluating limits often involves techniques such as substitution, factoring, or applying L'Hôpital's rule when dealing with indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they indicate the need for further analysis or manipulation of the expression to find the limit. Techniques such as algebraic simplification or L'Hôpital's rule are commonly used to resolve these forms.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the context of limits, rational functions often require careful analysis of their behavior as the variable approaches specific values, particularly when the denominator approaches zero. Understanding how to simplify these functions and identify removable discontinuities is essential for evaluating limits effectively.

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Textbook Question

Determine the following limits.

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

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).