Problem 8 - Single Variable Calculus

Question

Find the limit. $$ \lim _{t \rightarrow-3^{+}} \frac{t}{t+3} $$

Step-by-Step Solution

Answer

The short answer is: \(\lim_{t \rightarrow-3^{+}} \frac{t}{t+3} = -\infty\)

Step 1: Analyze the function as t approaches -3 from the right

Since \(t\) approaches -3 from the positive side, we can express it as (-3 + h), where \(h > 0\) and \(h\) is an infinitesimally small positive number. Plugging this expression into the function, we get: \(f(t) = \frac{-3 + h}{(-3 + h) + 3}\)

Step 2: Simplify the function

Simplifying the expression, we get: \(f(t) = \frac{-3 + h}{h}\)

Step 3: Find the limit as h approaches 0

Now, we'll find the limit as \(h\) approaches 0: \(\lim_{h \rightarrow 0}\frac{-3 + h}{h}\) As \(h\) approaches 0, the numerator becomes -3, and the denominator is still \(h\). Since h is still a positive number as it approaches 0, the limit would become: \(\lim_{h \rightarrow 0}\frac{-3}{h}\)

Step 4: Evaluate the limit

As the denominator approaches 0, the fraction becomes infinitely large and negative. Thus, the limit of the function as t approaches -3 from the right side is: \(\lim_{t \rightarrow-3^{+}} \frac{t}{t+3} = -\infty\)

Key Concepts

One-sided Limit

A one-sided limit refers to a limit where we only consider the values of a function as the variable approaches a certain point from one particular direction. This can either be from the left (negative direction) or from the right (positive direction). In this context, we are concerned with the one-sided limit from the right. This means that we only look at the behavior of the function as the variable approaches the given point from values greater than the point itself.
For example, when we consider the limit \(lim _{t \rightarrow-3^{+}} \frac{t}{t+3}\), we are actually interested in what happens to the function \(\frac{t}{t+3}\) as \(t\) gets closer and closer to -3, but only from values greater than -3. This approach restricts our focus to a specific direction, helping us understand the function's behavior in this context.

Approaching a Point from the Right

When we talk about approaching a point from the right, we are discussing how the values of a function change as the variable approaches a certain point by taking smaller and smaller steps from the right, or larger values. In our specific problem, \(t\) approaches -3 from the right. Thus, we express this as \(t = -3 + h\) where \(h\) is a positive number.
This method helps isolate the behavior of the function in this region, capturing the transition effect of the values as close as possible. By examining this behavior, we can understand how the function or expression uniquely reaches a certain value, or diverges to infinity, when coming specifically from the right side.

Infinitesimally Small Number

The concept of an infinitesimally small number is crucial when working with limits, particularly as it helps in capturing the 'approaching' aspect of limits. In mathematics, an infinitesimally small number is one that approaches zero, but never quite reaches it. In our exercise, \(h\), being such a number, represents an extremely small positive number as \(t\) approaches -3 from the right.
This theoretical construct lets us simplify expressions and analyze function behaviors in a precise manner. As \(h\) approaches zero, we observe significant changes like rapid increases or decreases in function values. That's why understanding infinitesimals is vital; they provide a practical way to focus on specific parts of the graph of the function where the insights are most relevant.