Problem 66 - University Calculus: Early Transcendentals

Question

Find the limits. $$\lim _{x \rightarrow 0^{-}} \sin x \cdot \ln x$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Function

We need to evaluate the limit \(\lim_{x \rightarrow 0^{-}} \sin x \cdot \ln x\). Here, "as \(x\) approaches zero from the left" can be tricky because the natural logarithm \(\ln x\) is undefined for non-positive numbers.

Step 2: Shift Perspective with Substitution

Let \(x = -t\), where \(t > 0\). As \(x \to 0^-\), \(t\) approaches zero from the right. Therefore, we rewrite our function in terms of \(t\) as \(-\sin(-t)\ln(-t) = \sin t \cdot \ln(-t)\).

Step 3: Analyze the Behavior of \(\ln(-t)\)

Notice for \(-t\) where \(t\to 0^+\), \(\ln(-t) = \ln(t) + i\pi\). The real part, \(\ln t\), goes to \(-\infty\). The imaginary part is constant, \(i\pi\).

Step 4: Assess the Combined Function

Recall that \(\sin t\) is bounded by \(-1\) and \(1\). Meanwhile, the real part of \(\ln t\) dominates, passing to \(-\infty\). Therefore, \(\sin t \cdot \ln t \rightarrow 0\) as \(t \rightarrow 0^+\).

Step 5: Evaluate the Limit

Hence, the original limit can be similarly evaluated. The real part tends to zero and the small changes from the \(i\pi\) constant don't affect the zero limit dominance. So the expression converges to zero reapplying the properties of limits.

Key Concepts

natural logarithm

The natural logarithm, often denoted as \( \ln x \), is a logarithmic function that has a base of \( e \), where \( e \approx 2.71828 \). This mathematical function is vital for dealing with exponential growth processes found in calculus and beyond. In terms of domain, the natural logarithm is only defined for positive values of \( x \). Hence, when we deal with functions such as \( \ln(-t) \), it requires some transformation to handle inputs associated with negative values.
To understand \( \ln x \) more comprehensively, remember that:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln e = 1 \) as \( e^1 = e \).
• As \( x \) increases towards infinity, \( \ln x \) also steadily increases.

When we examine the behavior of \( \ln(-t) \) when \( t \rightarrow 0^+ \), it becomes more complex due to its extension into the complex plane. Here, \( \ln(-t) \) can be written as \( \ln(t) + i\pi \), where the real component \( \ln(t) \) goes to \(-\infty \), representing a crucial behavior in solving limits.

sinusoidal functions

Sinusoidal functions such as \( \sin x \) and \( \cos x \) are essential trigonometric functions that describe periodic oscillations. In calculus limits, such functions often appear due to their oscillatory nature. The sine function, \( \sin x \), is particularly useful in limit problems because it is bounded; this means it never exceeds \( 1 \) or goes below \(-1\). This property can significantly aid in the evaluation of limits, especially when combined with growth functions like the logarithm.
Key characteristics of \( \sin x \) include:

• It has a fundamental periodic nature with a period of \( 2\pi \).
• Its value oscillates between \(-1\) and \(1\), inclusive.
• Maxima occur at \( \frac{\pi}{2} + 2n\pi \) and minima at \( \frac{3\pi}{2} + 2n\pi \), where \( n \) is an integer.

In the context of our limit problem, as \( \sin t \) remains bounded, the overall behavior of the product \( \sin t \cdot \ln t \) is largely dictated by the behavior of \( \ln t \). This bounded nature allows us to simplify and assert that if one part of the product approaches zero or \(-\infty\), it significantly impacts the entire expression.

limit evaluation techniques

In calculus, evaluating limits is a fundamental skill involving various techniques to understand the behavior of a function as it approaches a certain point. Analyzing the limit \( \lim_{x \rightarrow 0^{-}} \sin x \cdot \ln x \) requires thoughtful application of substitution and transformation.
A standard approach involves analyzing indeterminate forms or undefined expressions via substitution. In this context, by setting \( x = -t \), where \( t > 0 \), we simplify the expression manipulation while approaching \( t \rightarrow 0^{+} \). Here, \( \ln(-t) = \ln t + i\pi \) is split into real and imaginary parts.
Use these effective techniques when tackling similar limits:

• **Substitution**: Transform variables to aid in understanding limit behavior.
• **Splitting Functions**: Decompose complex expressions into manageable parts.
• **Identification of Dominating Behavior**: Determine which part of the expression dominates the limit's behavior, such as \( \ln t \) going to \(-\infty\) here.

Ultimately, evaluating our limit required understanding that the multiplicative dominance of \( \ln t \)'s negative growth led the entire expression towards zero, with \( \sin x \) bounded not impacting this result. Thus, the analyzed limit evaluated to zero, reaffirming the importance of these strategies in calculus.