Problem 20 - Thomas Calculus

Question

Use 1'Hopital's Rule to find the limits in Exercises \(7-26\). $$ \lim _{h \rightarrow 0} \frac{\sin (a+h)-\sin a}{h} $$

Step-by-Step Solution

Answer

The limit is \(\cos(a)\).

Step 1: Verify the Indeterminate Form

To use L'Hopital's Rule, we first check if the given limit results in an indeterminate form such as \(\frac{0}{0}\). Compute \(\sin(a+h)\) and \(\sin(a)\) as \(h\) approaches 0, yielding \(\sin(a) - \sin(a) = 0\) and the denominator \(h\), also approaching 0. Therefore, we have \(\frac{0}{0}\), an indeterminate form, allowing us to apply L'Hopital's Rule.

Step 2: Differentiate Numerator and Denominator

Apply L'Hopital's Rule, which involves differentiating the numerator and the denominator separately. The derivative of the numerator \(\sin(a+h) - \sin(a)\) with respect to \(h\) is \(\cos(a+h)\). The derivative of the denominator \(h\) is \(1\).

Step 3: Evaluate the Limit Using L'Hopital's Rule

Substitute the derivatives back into the limit expression. We have:\[\lim_{h \to 0} \frac{\cos(a+h)}{1}\]Now evaluate at \(h = 0\):\[\lim_{h \to 0} \cos(a+h) = \cos(a)\]

Step 4: Conclude the Limit Evaluation

Since we have found the limit by applying L'Hopital's Rule and evaluating at \(h = 0\), we conclude that:\[\lim_{h \to 0} \frac{\sin(a+h) - \sin(a)}{h} = \cos(a)\]

Key Concepts

Indeterminate Forms

Indeterminate forms often arise in calculus when we try to evaluate limits that initially seem undefined, such as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or \(0 \cdot \infty\). These forms are called "indeterminate" because they do not give enough information to determine the limit's actual value right away. When faced with an indeterminate form, calculators struggle to provide an answer directly. Enter L'Hopital's Rule, a handy tool for resolving such situations. It lets us differentiate the numerator and the denominator to find a limit analytically.In the exercise example, we observe the form \(\frac{0}{0}\) because both the numerator \(\sin(a+h) - \sin(a)\) and the denominator \(h\) approach zero as \(h\) approaches zero. Recognizing this form allows L'Hopital's Rule to be applied effectively.

Derivatives

Derivatives are a core concept in calculus, representing the rate of change of a function. They tell us how a function changes as its input changes, which is foundational for understanding limits and other advanced calculus topics. When using L'Hopital's Rule, deriving each part separately is crucial. In our solution, the derivative of the numerator \(\sin(a+h) - \sin(a)\) with respect to \(h\) is found using trigonometric differentiation rules. This yields \(\cos(a+h)\). When differentiating the denominator, \(h\), we get simply \(1\).

• The key is differentiating correctly; mistakes can lead to incorrect limits.
• Always double-check the common derivatives, especially for trigonometric functions, to ensure their correct application.

Trigonometric Limits

Trigonometric limits often appear in calculus, especially when dealing with sine, cosine, and tangent functions. These limits can sometimes be directly calculated, but often they're part of more complex compositions.In this exercise, we used the trigonometric identity and differentiation rules to tackle the limit of the sine function. After applying L'Hopital's Rule, we derive \(\cos(a+h)\) as the relevant trigonometric function for evaluating the limit.

• Understanding the behavior of trigonometric functions, such as their continuity and periodicity, aids in analyzing limits effectively.
• Recognize key trigonometric derivatives and memorize them to apply in similar problems.

Finally, by evaluating \(\cos(a+h)\) at \(h=0\), we find the limit simplifies directly to \(\cos(a)\), concluding our investigation.