Problem 67 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to 1/2}\ \textrm{arcsin}\ x$$

Step-by-Step Solution

Answer

The limit of \( \textrm{arcsin}\ x\) as \(x\) approaches \(1/2\) is \( \textrm{arcsin}(1/2)\) or approximately \(0.52359877\) in radians.

Step 1: Verify the function's continuity

In order to correctly use direct substitution, one needs to verify the function's continuity at \(x = 1/2\). The function \( \textrm{arcsin}\ x\) is indeed continuous at \(x = 1/2\), because the domain of \( \textrm{arcsin}\ x\) is \(-1 \leq x \leq 1\), which includes \(x = 1/2\).

Step 2: Apply direct substitution

Since the function is continuous at \(x = 1/2\), directly substitute \(1/2\) into the function to find the limit: \( \textrm{arcsin}(1/2)\).

Key Concepts

Direct Substitution Method

The direct substitution method is a fundamental approach in calculus for finding the limit of a function as a certain input value is approached. This technique is particularly useful when dealing with functions that are continuous at the point of interest.

To apply the direct substitution method, one simply replaces the variable in the equation with the limit value. If the resulting expression can be evaluated without any indeterminate forms such as 0/0 or ∞/∞, then the calculated value is the limit of the function at that point. It's important to confirm function continuity at the substitution point to avoid incorrect results.

Using direct substitution can simplify the process of finding limits and helps to avoid more complex methods like factoring, conjugate multiplication, or applying L'Hospital's rule. However, this method cannot always be used, especially in cases where direct substitution leads to an indeterminate form. In such scenarios, other techniques must be employed.

Arcsin Function

The arcsin function, also known as the inverse sine function, is written as \( \text{arcsin}(x) \), or sometimes \( \sin^{-1}(x) \). It answers the question, 'What angle in radians, when the sine function is applied, gives me x?'.

The domain for the arcsin function is \( -1 \leq x \leq 1 \) because these are the only values for which a real number angle can correspond in the unit circle. The range is \( -\frac{\pi}{2} \leq \text{arcsin}(x) \leq \frac{\pi}{2} \), representing angles from -90 to 90 degrees, or from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) radians.

For example, \( \text{arcsin}(1/2) \) is asking for the angle whose sine is 1/2. We know from trigonometry that this angle is 30 degrees or \( \frac{\pi}{6} \) radians, making it a straightforward application of the arcsin function.

Function Continuity

Function continuity is a property that describes whether a function is continuous at a certain point or over an interval. A continuous function is one where small changes in the input value result in small changes in the output value, without any sudden jumps or breaks in the graph.

Mathematically, a function \( f(x) \) is continuous at a point \( a \) if three conditions are met: \( \lim_{x \to a^-}f(x) = \lim_{x \to a^+}f(x) = f(a) \), meaning the left-hand limit equals the right-hand limit and also equals the function's value at that point.

Continuity at a point ensures that direct substitution is valid for finding limits, as seen in the provided exercise. If a function is not continuous at the point, the limit may exist, but finding it requires other methods, as simply plugging in the number might not give the correct result.

Limit of a Function

In calculus, the limit of a function is a fundamental concept that describes the behavior of that function as it approaches a specific input value, commonly denoted as \( x \). The limit does not always equal the function's actual value at that point, particularly if the function is not continuous there.

Mathematically, the notation \( \lim_{x \to a} f(x) \) represents the limit of the function \( f(x) \) as \( x \) approaches \( a \) from either side. This concept is critical when analyzing points of discontinuity, end behavior, and approaching infinity.

Limits are used to define many other concepts in calculus, such as derivatives and integrals, and are essential in understanding change and motion. Mastery of limits lays the foundation for further study in calculus and its diverse applications in science, engineering, economics, and beyond.