Problem 57 - Thomas Calculus

Question

Limit of a quotient Suppose that functions \(g(t)\) and \(h(t)\) are defined for all values of \(t\) and \(g(0)=h(0)=0 .\) Can \(\lim _{t \rightarrow 0}(g(t)) /(h(t))\) exist? If it does exist, must it equal zero? Give reasons for your answers.

Step-by-Step Solution

Answer

The limit can exist and be non-zero if \(\lim_{t \rightarrow 0} \frac{g'(t)}{h'(t)}\) is finite.

Step 1: Define the problem

We need to determine whether the limit \(\lim_{t \rightarrow 0} \frac{g(t)}{h(t)}\) exists given that both \(g(t)\) and \(h(t)\) are functions defined for all \(t\) and \(g(0) = h(0) = 0\). We also need to examine if the limit, if it exists, must be zero.

Step 2: Use L'Hôpital's Rule

L'Hôpital's Rule is applicable when both the numerator and the denominator approach zero as \(t \rightarrow 0\). To apply this rule, differentiate the numerator and denominator:\(\lim_{t \rightarrow 0} \frac{g(t)}{h(t)} = \lim_{t \rightarrow 0} \frac{g'(t)}{h'(t)}\) provided that this limit exists.

Step 3: Existence of the limit

The existence of \(\lim_{t \rightarrow 0} \frac{g(t)}{h(t)}\) depends on \(\lim_{t \rightarrow 0} \frac{g'(t)}{h'(t)}\). If the latter limit exists and is finite, then the original limit exists.

Step 4: Consider if the limit must be zero

The limit will not necessarily be zero. If \(\lim_{t \rightarrow 0} \frac{g'(t)}{h'(t)}\) exists and is non-zero, then \(\lim_{t \rightarrow 0} \frac{g(t)}{h(t)}\) can be any real number or even infinity.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a very powerful tool in calculus that helps determine limits of indeterminate forms. It applies when both the numerator and the denominator of a fraction approach 0 or infinity as the variable approaches a certain value. In these cases, directly computing the limit can be difficult or impossible, so we use L'Hôpital's Rule to make our calculations easier.

Here is how it works:

• First, confirm that both the numerator and the denominator result in an indeterminate form like 0/0 or ∞/∞ as the variable approaches some value.
• Next, differentiate both the numerator and the denominator separately.
• Then, find the limit of the new fraction, \(lim_{t \to a} \frac{f'(t)}{g'(t)}\). If this limit exists, it equals the original fraction's limit.
• You can apply L'Hôpital's Rule multiple times if needed until the limit is determined.

Always verify whether the conditions for L'Hôpital's Rule are met, which include the differentiability of the original functions over an interval around the point of interest (except possibly at the point itself).

Indeterminate Forms

Indeterminate forms occur when evaluating a limit results in an unclear or undefined form, such as 0/0 or ∞/∞. These forms do not immediately provide information about the limit's value because they suggest multiple possibilities.

A few common indeterminate forms include:

• 0/0: A limit that appears to result in a fraction with a zero numerator and denominator.
• ∞/∞: A fraction where both parts trend towards infinity.
• There are other forms too, such as 0⋅∞, ∞−∞, and 1^∞, which also need special techniques for evaluation.

Methods such as algebraic manipulation, factorization, or L'Hôpital's Rule can be employed to resolve these forms. Each situation may require a different approach based on the specific behavior of the functions involved. Identifying indeterminate forms is crucial because it alerts us that a straightforward substitution is not enough, and additional algebraic or calculus-based strategies need to be employed.

Differentiable Functions

A differentiable function is one that has a derivative at every point in its domain. This means we can calculate the slope of the tangent line at any point on the function's curve, which is central to finding derivatives.

Key features of differentiable functions include:

• Continuity: If a function is differentiable at a point, it must also be continuous at that point. However, not all continuous functions are differentiable.
• Smoothness: Differentiability implies the curve of the function is smooth (no sharp corners or cusps).
• Limit existence: The derivative itself is a limit, specifically the limit of the difference quotient as the interval approaches zero. This concept ties back to calculating more complex limits like those handled by L'Hôpital's Rule.

Differentiability is a critical concept because it guarantees that we can apply calculus techniques like differentiation, which in turn allows the use of tools like L'Hôpital's Rule and helps solve certain limit problems. Knowing whether a function is differentiable guides us in our analysis of calculus problems, including those involving indeterminate forms.