Question

Evaluate the following limits. \(\lim _{t \rightarrow-2}\left(t^{2}+5 t+7\right)\)

Short answer

Question: Evaluate the limit: \(\lim _{t \rightarrow -2} (t^2+5t+7)\) Answer: The limit is equal to 1: \(\lim _{t \rightarrow -2} (t^2+5t+7) = 1\).

Step-by-step solution

Identify the given limit

We are given the limit \(\lim _{t \rightarrow -2} (t^2+5t+7)\) and we need to evaluate it.

Substitute the given value of t

Since this is a polynomial function, we can simply substitute the value of \(t\) directly to find the limit. Replace \(t\) with \(-2\) in the expression: \((-2)^2 + 5(-2) + 7\)

Simplify the expression

Calculate the expression by following the order of operations (PEMDAS): \(4 - 10 + 7\)

Compute the answer

Add and subtract the values in step 3: \(4 - 10 + 7 = -6 + 7 = 1\) The limit is equal to 1: \(\lim _{t \rightarrow -2} (t^2+5t+7) = 1\).

Key concepts

Understanding Limits

In calculus, limits are a fundamental concept that describe how a function behaves as its input approaches a particular value. They help us understand the behavior of functions at points where they may not be directly defined. This is particularly useful in understanding points of discontinuity or at boundaries. When we talk about evaluating a limit like \(\lim_{t \rightarrow -2} (t^2 + 5t + 7)\), we're expressing the idea that we're interested in what value the function \(t^2 + 5t + 7\) gets closer to as \(t\) approaches \(-2\). Limits are the building blocks of derivatives and integrals, which are major topics in calculus. They help to define these concepts rigorously and are used to solve various mathematical problems related to change and areas.

Introduction to Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of variables with coefficients. The general form of a polynomial function can be expressed as \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(n\) is a non-negative integer, and the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are real numbers. A polynomial function of degree 2, like the one in our exercise, is called a quadratic function. The quadratic function in our problem, \(t^2 + 5t + 7\), will have a smooth and continuous curve. This attribute of polynomials means that they are particularly nice to work with when evaluating limits, because they don't have any jumps or holes. You can usually evaluate the limit of a polynomial by simply substituting the value of the variable directly into the function, as demonstrated in the exercise.

Evaluating Limits of Polynomial Functions

Evaluating the limits of polynomial functions is generally a straightforward process compared to other types of functions. When the expression is a polynomial, like \(t^2 + 5t + 7\), you can evaluate the limit by direct substitution, which means you replace the variable with the value it approaches. For the given problem \(\lim_{t \rightarrow -2} (t^2 + 5t + 7)\):

• Substitute \(t = -2\) into the polynomial to get \((-2)^2 + 5(-2) + 7\).
• Simplify by following the order of operations: \(4 - 10 + 7\).
• Compute the final result: \(1\).

The simplicity here is because polynomial functions are continuous everywhere. This means limits can be found by substitution without worrying about discontinuities or undefined points. This makes polynomial limits easy to handle in calculus.