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Chapter 1: Problem 34

Find the limits. $$ \lim _{x \rightarrow 0} \frac{x}{\cos \left(\frac{1}{2} \pi-x\right)} $$

Short Answer

Expert verified
The limit is 1.

Step by step solution

01

Apply Trigonometric Identity

First, note that the expression inside the cosine function can be simplified using the trigonometric identity: \( \cos \left(\frac{\pi}{2} - x\right) = \sin(x) \). Substitute this identity into the limit expression to get \( \lim_{x \to 0} \frac{x}{\sin(x)} \).
02

Use Standard Limit

Recognize that \( \lim_{x \to 0} \frac{x}{\sin(x)} = 1 \), which is a standard trigonometric limit. This allows us to directly evaluate the limit of the expression from Step 1 to find the result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are essential tools in calculus, especially when dealing with limits involving trigonometric functions. These identities allow us to rewrite expressions in simpler or more recognizable forms. In this exercise, we used the identity for cosine of an angle minus a variable: \( \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \).
This particular identity is known as the co-function identity, and it transforms a cosine function involving a subtraction into a sine function involving the variable directly.

Such conversions are helpful because they can simplify limit problems by making the expression easier to handle or by allowing us to use known limits.
  • By recognizing the presence of trigonometric identities, you can transform complex trigonometric expressions into more manageable forms.
  • This helps in applying other techniques or known results effectively.
Standard Trigonometric Limits
Standard trigonometric limits are specific limit results that are frequently used in calculus problems, particularly those involving trigonometric functions. A key example is the limit \( \lim_{x \to 0} \frac{x}{\sin(x)} = 1 \).
This result stems from the behavior of the sine function near zero, where \( \sin(x) \approx x \) for small values of \( x \).

This approximation translates into the standard limit mentioned above, which is commonly memorized and used as a tool to evaluate limits quickly.
  • These limits are foundational since they often appear in problems or as steps in simplifying more complex expressions.
  • Memorizing them allows for faster and more intuitive problem-solving.
Understanding these concepts can help streamline calculations in problems similar to the one in this exercise.
Limit Evaluation Techniques
Limit evaluation techniques are strategies and methods used to find the limit of a function as a variable approaches a particular value. One of the common techniques is substitution, as seen in this exercise where a trigonometric identity transformed the initial expression.
Afterward, recognizing and applying a standard trigonometric limit can quickly solve the problem.

Other common techniques include:
  • Factoring: Simplifying the expression by factoring out the common terms.
  • Rationalization: Multiplying by a conjugate to remove square roots or complex expressions in the denominator.
  • L'H么pital's Rule: Used when the limit results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), allowing differentiation of numerator and denominator.
Mastering these techniques makes limit evaluation more accessible and prepares you for different types of calculus problems. By learning these strategies, you can tackle a variety of limits efficiently, as demonstrated with different methods.

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Most popular questions from this chapter

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves \(y=f(x)\) and \(y=g(x)\) are asymptotic as \(x \rightarrow+\infty\) provided$$\lim _{x \rightarrow+\infty}[f(x)-g(x)]=0$$and are asymptotic as \(x \rightarrow-\infty\) provided$$\lim _{x \rightarrow-\infty}[f(x)-g(x)]=0$$In these exercises, determine a simpler function \(g(x)\) such that \(y=f(x)\) is asymptotic to \(y=g(x)\) as \(x\rightarrow+\infty\) or \(x \rightarrow-\infty\). Use a graphing utility to generate the graphs of \(y=f(x)\) and \(y=g(x)\) and identify all verticalasymptotes. $$ f(x)=\frac{-x^{3}+3 x^{2}+x-1}{x-3} $$

Show that the equation \(x^{3}+x^{2}-2 x=1\) has at least one solution in the interval \([-1,1]\).

True-False Determine whether the statement is true or false. Explain your answer. If \(\lim _{x \rightarrow a} f(x)\) and \(\lim _{x \rightarrow a} g(x)\) both exist and are equal, then \(\lim _{x \rightarrow a}[f(x) / g(x)]=1\).

In the study of falling objects near the surface of the Earth, the acceleration \(g\) due to gravity is commonly taken to be a constant \(9.8 \mathrm{~m} / \mathrm{s}^{2}\). However, the elliptical shape of the Earth and other factors cause variations in this value that depend on latitude. The following formula, known as the World Geodetic System 1984 (WGS 84) Ellipsoidal Gravity Formula, is used to predict the value of \(g\) at a latitude of \(\phi\) degrees (either north or south of the equator): \(g=9.7803253359 \frac{1+0.0019318526461 \sin ^{2} \phi}{\sqrt{1-0.0066943799901 \sin ^{2} \phi}} \mathrm{m} / \mathrm{s}^{2}\) (a) Use a graphing utility to graph the curve \(y=g(\phi)\) for \(0^{\circ} \leq \phi \leq 90^{\circ}\). What do the values of \(g\) at \(\phi=0^{\circ}\) and at \(\phi=90^{\circ}\) tell you about the WGS 84 ellipsoid model for the Earth? (b) Show that \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) somewhere between latitudes of \(38^{\circ}\) and \(39^{\circ}\).

Writing In your own words, explain the practical value of the Squeezing Theorem.
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