Determine the following limits.

lim u→0^+ u − 1 / sin u

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Identify the limit expression: $ \lim_{{u \to 0^+}} \frac{u - 1}{\sin u} $.

Recognize that direct substitution of $ u = 0 $ results in an indeterminate form $ \frac{-1}{0} $, which suggests the need for further analysis.

Consider using L'Hôpital's Rule, which applies to limits of the form $ \frac{0}{0} $ or $ \frac{\infty}{\infty} $. However, since this is not directly applicable here, explore other methods.

Analyze the behavior of $ \sin u $ as $ u \to 0^+ $. Recall that $ \sin u \approx u $ for small $ u $, which can help simplify the expression.

Rewrite the limit using the approximation: $ \lim_{{u \to 0^+}} \frac{u - 1}{u} $ and evaluate this new expression to find the limit.

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Key Concepts

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

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as u approaches 0 from the positive side, which is crucial for evaluating the expression given.

Sine Function

The sine function, denoted as sin(u), is a periodic function that oscillates between -1 and 1. It is essential in calculus for analyzing limits, especially when dealing with small angles. As u approaches 0, sin(u) can be approximated by its Taylor series expansion, which is useful for simplifying expressions involving limits.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(u)/g(u) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in the given limit problem, where direct substitution leads to an indeterminate form.

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Evaluate each limit and justify your answer.

lim x→4 √x^3−2x^2−8x / x−4

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Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→5 1/x^2=1/25

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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).