Problem 24 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{t \rightarrow-2}\left(t^{2}+5 t+7\right)$$

Step-by-Step Solution

Answer

Answer: The limit of the function as \(t\) approaches \(-2\) is 1.

Step 1: Substitute the value of \(t\) in the function

Since we need to find the limit as \(t\) approaches \(-2\), we will substitute \(t = -2\) into the function: $$f(t) = t^2 + 5t + 7$$ $$f(-2) = (-2)^2 + 5(-2) + 7$$

Step 2: Simplify the expression

Now, we just need to simplify the expression to find the value of the function at \(t = -2\): $$f(-2) = 4 + (-10) + 7$$ $$f(-2) = 4 - 10 + 7$$ $$f(-2) = 1$$

Step 3: State the limit

Since the value of the function at \(t = -2\) is \(1\), we can say that the limit of the function as \(t\) approaches \(-2\) is: $$\lim_{t \rightarrow -2} \left(t^2 + 5t + 7\right) = 1$$

Key Concepts

Limits

Limits are fundamental to calculus and help us understand the behavior of functions as they approach specific points. When examining limits, we investigate what value a function approaches as the input gets closer to a certain point. This is essential in understanding continuity and the general behavior of a function.
A limit is expressed as \( \lim_{x \rightarrow a} f(x) \), where \( f(x) \) represents the function and \( a \) is the point of interest. The goal is to determine the value that \( f(x) \) gets closer to, but not necessarily attains, as \( x \) nears \( a \).
It is worth noting that limits can exist, be infinite, or not exist at all depending on the function behavior around the point in question.

Polynomial Functions

Polynomial functions are expressions composed of variables raised to various powers, combined using addition, subtraction, and multiplication. They are among the most common function types in mathematics due to their simplicity and versatility.
For example, in the expression \( t^2 + 5t + 7 \), we have a polynomial of degree 2, known as a quadratic polynomial, because it includes a term with \( t^2 \).

• Constants: The number 7 in this polynomial is a constant term.
• Linear Term: The term \( 5t \) is linear, involving \( t \) to the power of 1.
• Quadratic Term: The term \( t^2 \) is quadratic, involving \( t \) to the power of 2.

Polynomials are continuous and defined everywhere for real numbers, making them simple to evaluate for limits.

Limit Evaluation

Evaluating limits involves calculating the value a function approaches as the input nears a certain point. There are several methods, such as substitution, factoring, or using special limit rules. For polynomial functions, direct substitution is commonly used because of their continuity.
In the exercise provided, we use direct substitution to evaluate the limit of the polynomial \( t^2 + 5t + 7 \) as \( t \) approaches \(-2\).
By substituting \( t = -2 \) directly into the polynomial, we calculate the resulting value, ensuring that the function behaves predictably near the point of interest. This method is efficient for polynomials because they do not have any discontinuities to worry about.

Substitution Method

The substitution method is a straightforward technique for finding limits, especially useful for polynomials and other continuous functions. Here's how it works:

1. Identify the point \( a \) that \( t \) is approaching in the limit expression.
2. Substitute \( t = a \) directly into the function \( f(t) \).
3. Simplify the expression to find the limit value.

In the exercise example, the substitution method was used by replacing \( t \) with \(-2\) in the polynomial \( t^2 + 5t + 7 \). The result \( f(-2) = 1 \) informs us that as \( t \) approaches \(-2\), the function approaches the value 1. This technique leverages the continuity of polynomial functions, which makes the evaluation straightforward and reliable.