Problem 29 - Calculus for Biology and Medicine

Question

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow \pi / 3} \sin \left(\frac{x}{2}\right) $$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Understand the Limit

We need to find the limit of the function \( \sin\left(\frac{x}{2}\right) \) as \( x \) approaches \( \frac{\pi}{3} \). This involves evaluating the function at a value where the input to the sine function, \( \frac{x}{2} \), becomes \( \frac{\pi}{6} \).

Step 2: Substitute the Limit Point

Substitute \( x = \frac{\pi}{3} \) into the function, so it becomes \( \sin\left(\frac{\pi}{3}/2\right) \). This simplifies to \( \sin\left(\frac{\pi}{6}\right) \).

Step 3: Evaluate the Sine Function

Calculate \( \sin\left(\frac{\pi}{6}\right) \). From trigonometry, we know that \( \sin\left(\frac{\pi}{6}\right) \) is equal to \( \frac{1}{2} \).

Step 4: Conclusion on the Limit

With the calculations above, we find that the limit of \( \sin\left(\frac{x}{2}\right) \) as \( x \) approaches \( \frac{\pi}{3} \) is \( \frac{1}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics for modeling cyclical or periodic behaviors. They include sine, cosine, and tangent, among others. These functions relate to angles and ratios derived from a right triangle. For example, the sine function (often written as \( \sin \)) gives the ratio of the opposite side to the hypotenuse length in a right-angled triangle. Curious about the unit circle? Trig functions also describe points on this circle:

• The circle’s radius is 1.
• The angle's cosine and sine correspond to x and y coordinates of a point.

For example, when evaluating \( \sin\left(\frac{\pi}{6}\right) \), we look at a specific angle’s behavior on this circle, finding that it results in \( \frac{1}{2} \). This knowledge becomes key when we apply these concepts to solve problems involving limits and substitutions.

Substitution Method

The substitution method is a straightforward yet powerful technique used in evaluating limits. This approach involves replacing the variable of the limit with a given value and simplifying. When faced with \( \lim_{x \rightarrow \pi/3} \sin \left(\frac{x}{2}\right) \), substituting \( x = \pi/3 \) is our first task.
This substitution simplifies the function from \( \sin \left(\frac{x}{2}\right) \) to \( \sin \left(\frac{\pi/6}\right) \). This simplification is crucial since it transforms the expression into something immediately solvable using basic trigonometric identities.
The power of substitution lies in its ability to simplify expressions and transform limits into values, reducing potential confusion when approaching troublesome algebraic or trigonometric forms.

Limits Evaluation

Evaluating limits involves understanding a function's behavior as the variable approaches a particular value. It's a core concept in calculus, helping describe continuity and predict function trends.
When assessing \( \lim_{x \rightarrow \pi/3} \sin \left(\frac{x}{2}\right) \), we aim to find the output of the sine function as its input nears \( \pi/6 \). We substitute directly and then evaluate \( \sin(\frac{\pi}{6}) = \frac{1}{2} \).
This method illustrates how combining direct substitution and fundamental trigonometric values efficiently solves limit problems. While often seeming complex, systematically applying rules and recognizing patterns makes limit evaluation more intuitive and approachable for students.