Problem 27 - Precalculus with Limits

Question

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{t\to \infty} \left( \dfrac{1}{3t^2} -\dfrac{5t}{t+2} \right) \\]

Step-by-Step Solution

Answer

The limit of the given expression as \(t\) approaches infinity is \(-\frac{5}{3}\).

Step 1: Simplify the Expression

Let's start by simplifying \(\frac{1}{3t^2} -\frac{5t}{t+2}\). It saves us a lot of time to write the function in more simplified form. Write the function as: \(-\frac{5t^2+10t-3}{3t^2(t+2)}\).

Step 2: Divide Each Term by \(t^2\)

Now, divide each term in the numerator and the denominator of the fraction by \(t^2\), getting \(-\frac{5 + \frac{10}{t}-\frac{3}{t^2}}{3(t+\frac{2}{t})}\). This step will help to take the limit as \(t\) approaches infinity.

Step 3: Take the Limit

Taking the limit as \(t\) goes to infinity, the terms \(\frac{10}{t}\), \(\frac{3}{t^2}\), and \(\frac{2}{t}\) tend towards 0, as they are inversely proportional to \(t\). Therefore, the limit is \(-\frac{5}{3}\).

Step 4: Check with Graphing Utility

It's often useful to verify the results graphically. In this case, if you graph the function, you will see it converging towards \(-\frac{5}{3}\) as \(t\) goes to infinity.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a critical first step when dealing with complex limits in calculus. It's about rewriting an expression in a form that is easier to work with. For this exercise, we are tasked with simplifying the expression \(\frac{1}{3t^2} - \frac{5t}{t+2} \). The goal here is to combine terms into a single expression.
By observing the original expression, the key is to eliminate fractions and find a common structure. For this, we combined both parts of the expression under a common denominator: \(3t^2(t + 2)\).

• Convert \(\frac{1}{3t^2}\) and \(-\frac{5t}{t + 2}\) into a single fraction \(-\frac{5t^2 + 10t - 3}{3t^2(t + 2)}\).
• We achieve this by finding common factors and minus signs in the respective numerators and denominators to simplify.

This simplified version not only makes the math manageable but also highlights dominant terms that dictate the behavior at infinity.

Evaluating Infinite Limits

Evaluating infinite limits involves analyzing what happens to a function as the variable approaches infinity. Here, the limit is represented as \(\lim_{t\to \infty} \left( \dfrac{1}{3t^2} - \dfrac{5t}{t+2} \right)\).
To evaluate the limit, we start by dividing each term in both the numerator and the denominator by \(t^2\), the highest power of \(t\) present. This is done to facilitate the limit evaluation as it approaches infinity:

• In the term \(\frac{1}{3t^2}\), dividing by \(t^2\) makes it simplify directly.
• For \(-\frac{5t}{t+2}\), normalizing by \(t^2\) results in \(-\frac{5 + \frac{10}{t} - \frac{3}{t^2}}{3(t+\frac{2}{t})}\).

As \(t\) increases without bound, \(\frac{10}{t}\), \(\frac{3}{t^2}\), and \(\frac{2}{t}\) collapse to zero, simplifying our expression to just \(-\frac{5}{3}\). This leftover fraction is indeed the limit as \(t\to \infty\). It emphasizes that terms with higher powers in the denominator vanish, overshadowed by dominant constant terms.

Graphical Verification of Limits

Graphical verification of limits offers a visual confirmation of our analytic results by using graphing tools to display the behavior of expressions as variables extend toward infinity.
Using a graphing utility, plot the simplified function \(-\frac{5t^2+10t-3}{3t^2(t+2)}\). As \(t\) grows larger, observe how the curve shows its asymptotic behavior converging towards \(-\frac{5}{3}\).

• Open your graphing utility and input the given function.
• Focus on the behavior of the graph as \(t\) tends to infinity.
• Notice the horizontal asymptote at \(-\frac{5}{3}\), which supports our calculated limit.

This exercise underscores the power of graphical analysis in confirming algebraic findings, combining the visual tool with calculus concepts to build a more robust understanding. Such graphics help students intuitively grasp why the calculated limit forms as it does, providing a multi-faceted approach to observing limit behavior.