Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{t \rightarrow-3^{+}} \frac{t^{2}-9}{t+3}\)

Step-by-Step Solution

Answer

The limit is -6.

Step 1: Understand the Problem

We need to find the right-hand limit of the function \( f(t) = \frac{t^2 - 9}{t+3} \) as \( t \) approaches \(-3\) from the positive side.

Step 2: Simplify the Expression

Notice that \( t^2 - 9 \) is a difference of squares, which can be factored as \((t+3)(t-3)\). This simplifies the expression to \( f(t) = \frac{(t+3)(t-3)}{t+3} \).

Step 3: Simplify Further by Canceling Terms

Since \( t+3 eq 0 \) as \( t \to -3^+ \), we can cancel the \( t+3 \) terms from the numerator and denominator. The simplified expression is \( f(t) = t - 3 \).

Step 4: Evaluate the Limit

Now, substitute \( t = -3 \) into the simplified expression \( t - 3 \). This gives \( -3 - 3 = -6 \).

Step 5: Conclusion on the Limit

Since \( t \) approaches \(-3\) from the right, \( f(t) = t - 3 \) remains defined and continuous around this point. Thus, the limit is \(-6\).

Key Concepts

Right-hand Limit

A right-hand limit refers to the value that a function approaches as the input approaches a specific point from the right side. In mathematics, this is often notated as \( \lim_{x \to a^+} f(x) \). This notation specifically indicates that our focus is on what happens as the variable \( x \) gets close to \( a \), while \( x \) is greater than \( a \).
When solving for right-hand limits, we observe as the input gets increasingly close to the target number but remains slightly higher than that target. This approach is particularly useful when dealing with piecewise functions or points of discontinuity, where the behavior of the function might differ when approaching from different sides.
For example, in our problem, we are asked to find the limit \( \lim _{t \rightarrow -3^{+}} \frac{t^{2}-9}{t+3} \). The notation \( -3^+ \) means we examine how the function behaves as \( t \) comes from numbers greater than \(-3\). This helps us determine the limit safely without crossing the point where the function may not be initially defined.

Difference of Squares

The difference of squares is a useful algebraic identity that simplifies expressions where one square number is subtracted from another. This identity is expressed as \( a^2 - b^2 = (a+b)(a-b) \). The application of this identity can make complex algebraic problems far easier to handle, particularly simplifying rational expressions.
In our example, the expression \( t^2 - 9 \) can be factored into \( (t+3)(t-3) \).

• Here, \( t^2 \) is the square of \( t \) and \( 9 \) is the square of \( 3 \).
• By recognizing this as a difference of squares, we can split \( t^2 - 9 \) into \((t+3)(t-3)\), simplifying our original problem.

This transformation is crucial when dealing with limits, as it often allows the cancellation of terms that might otherwise cause division by zero.
By simplifying the expression through factoring, it's easier to understand the behavior of the function around problematic points, like \(-3\) in our exercise.

Function Continuity

Function continuity at a point refers to the state when three conditions are satisfied:

• The function is defined at the point.
• The limit of the function as it approaches the point exists.
• The value of the function at that point is exactly equal to the limit at that point.

Having a continuous function means there is no sudden jump or break in the graph at that point. Functions that meet the above conditions at every point in their domain are considered continuous everywhere.
In the exercise, once we simplified \( \frac{t^2 - 9}{t+3} \) to \( t - 3 \), we concluded that as \( t \rightarrow -3^{+} \), the limit is \(-6\). The function is continuous because as we approach \(-3\) from the right side, we found \( f(t) = t - 3 \) does not disrupt or "jump" when \( t \) is slightly greater than \(-3\). Thus, it confirms continuity in that interval, making the evaluation of the limit straightforward. Continuity helps us verify the behavior of functions in their neighborhood, ensuring we can safely substitute values in the simplified expressions without unexpectedly hitting an undefined term.