Problem 39 - Calculus for Biology and Medicine

Question

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-1}\left(x^{3}+7 x-1\right) $$

Step-by-Step Solution

Answer

The limit is -9.

Step 1: Identify the Limit Expression

The limit expression given is \( \lim_{x \rightarrow -1} (x^{3} + 7x - 1) \). This expression indicates that we are trying to find the value of the function \( x^{3} + 7x - 1 \) as \( x \) approaches \( -1 \).

Step 2: Plug in the Value

We can straightforwardly evaluate this limit by directly substituting \( x = -1 \) into the polynomial since polynomials are continuous functions.\( (-1)^3 + 7(-1) - 1 \).

Step 3: Calculate the Cubic Term

Calculate \( (-1)^3 \):\(-1 \times -1 \times -1 = -1 \).

Step 4: Calculate the Linear Term

Calculate the linear term \( 7(-1) \):\(-7 \).

Step 5: Add the Constant Term

Consider the constant term \(-1\).

Step 6: Sum All Terms

Now, sum all these calculated terms together: \( -1 + (-7) + (-1) \). This results in \(-9\).

Step 7: Final Step: State the Limit Result

By performing the substitution and calculations, the limit evaluates to \(-9\). Therefore, we can conclude that \( \lim_{x \rightarrow -1} (x^{3} + 7x - 1) = -9 \).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that comprises variables raised to a whole number power, which are multiplied by coefficients and added together. These functions are characterized by coefficients and exponents that are non-negative integers. Commonly encountered in calculus are simple polynomials such as linear functions, quadratic, cubic, and higher degrees.

• A general example of a polynomial is of the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where each \( a_i \) represents a coefficient and \( n \) indicates the degree of the polynomial.
• The leading term in the polynomial is the one with the highest power of \( x \).
• The behavior of polynomial functions is smooth and continuous, which makes them relatively easier to handle when it comes to calculus.

In our exercise, the function \( x^3 + 7x - 1 \) is a cubic polynomial. Its polynomial structure suggests it is continuous and defined for all real numbers. This continuity is crucial for evaluating limits directly.

Continuity

Continuity is a fundamental concept in calculus, which ensures there are no sudden jumps or breaks in a function. A function is said to be continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.

• A mathematical function \( f(x) \) is continuous at \( x = c \) if \( \lim_{x \to c} f(x) = f(c) \).
• For polynomial functions, they are inherently continuous across their domains, meaning there are no breaks or points of discontinuity.
• This property makes evaluating limits more straightforward, as you can directly substitute the point into the function without concern for undefined points or division by zero.

In our example, the continuity of the polynomial \( x^3 + 7x - 1 \) allows us to evaluate the limit by simply substituting \( x = -1 \) without needing any special adjustments or considerations.

Limit Laws

Limit laws are a set of rules that facilitate easier calculation of limits, making the process straightforward and reliable. These laws can apply to various arithmetic operations performed on limits, including addition, subtraction, multiplication, and division.

• The Sum/Difference Law states that the limit of a sum or difference of functions is simply the sum or difference of their limits: \( \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \).
• The Constant Multiple Law suggests that the limit of a constant multiplied by a function's limit is equal to the constant multiplied by the limit: \( \lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x) \).
• The Power Law indicates that the limit of a function raised to a power is the limit of the function raised to that power: \( \lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n \).

In our exercise, leveraging these laws allows us to break down the polynomial expression \( x^3 + 7x - 1 \) into simpler parts. The continuity of polynomials, as noted, makes these laws particularly powerful because they enable us to substitute \( x = -1 \) directly and simply sum the effects of each term to find the final limit.