Business Calculus Chapter 1 Study Guide – Limits, Continuity, and Derivatives

Q1. Find the limit by completing the input-output tables as $x \to -7^+$ for $f(x) = \frac{x^2 + 7x}{x^2 - 49}$.

Background

Topic: Limits and One-Sided Limits

This question tests your understanding of how to estimate limits numerically using tables, especially as $x$ approaches a value from the right ($x \to -7^+$).

Key Terms and Formulas:

• Limit: The value that $f(x)$ approaches as $x$ gets closer to a specific point.

• One-sided limit: The value $f(x)$ approaches as $x$ approaches from one side (right or left).

• Function: $f(x) = \frac{x^2 + 7x}{x^2 - 49}$

Step-by-Step Guidance

1. Identify the function and the direction of approach: $x \to -7^+$ means you are approaching $-7$ from values greater than $-7$ (e.g., $-6.9, -6.99, -6.999$).

2. Substitute values slightly greater than $-7$ into $f(x)$ to fill out the table. For example, calculate $f(-6.9)$, $f(-6.99)$, and $f(-6.999)$.

3. For each value, compute the numerator and denominator separately, then divide to find $f(x)$.

4. Observe the pattern in the output values as $x$ gets closer to $-7$ from the right. What value does $f(x)$ seem to approach?

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Q2. Find $\lim_{x \to -7^+} f(x)$, $\lim_{x \to -7^-} f(x)$, and $\lim_{x \to -7} f(x)$, if each exists.

Background

Topic: One-Sided and Two-Sided Limits

This question asks you to determine the left-hand limit, right-hand limit, and overall limit at $x = -7$ for the given function.

Key Terms and Formulas:

• Left-hand limit: $\lim_{x \to -7^-} f(x)$

• Right-hand limit: $\lim_{x \to -7^+} f(x)$

• Two-sided limit: $\lim_{x \to -7} f(x)$ exists only if both one-sided limits are equal.

Step-by-Step Guidance

1. Recall your results from the table in Q1 for the right-hand limit ($x \to -7^+$).

2. Repeat the process for values slightly less than $-7$ (e.g., $-7.1, -7.01, -7.001$) to estimate the left-hand limit.

3. Compare the left-hand and right-hand limits. If they are equal, the two-sided limit exists and equals that value; if not, the two-sided limit does not exist.

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Q3. Find the limit by graphing the function $f(x) = \frac{x^2 + 7x}{x^2 - 49}$.

Background

Topic: Graphical Limits

This question tests your ability to estimate limits by analyzing the graph of a function near a specific point.

Key Terms and Formulas:

• Graphical limit: The $y$-value the function approaches as $x$ gets close to a certain value, as seen on the graph.

Step-by-Step Guidance

1. Sketch or use a graphing tool to plot $f(x)$ near $x = -7$.

2. Observe the behavior of the function as $x$ approaches $-7$ from both sides.

3. Identify any vertical asymptotes, holes, or jumps at $x = -7$.

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Q4. Find the limit algebraically for $f(x) = \frac{x^2 + 7x}{x^2 - 49}$ as $x \to -7$. Show your work.

Background

Topic: Algebraic Limits and Simplification

This question tests your ability to simplify rational functions and evaluate limits algebraically, especially when direct substitution leads to an indeterminate form.

Key Terms and Formulas:

• Indeterminate form: $\frac{0}{0}$, which requires algebraic manipulation.

• Factoring: Factor numerator and denominator to simplify the expression.

Step-by-Step Guidance

1. Substitute $x = -7$ into the function to check if you get an indeterminate form.

2. If you get $\frac{0}{0}$, factor both the numerator and denominator to see if you can cancel common terms.

3. Simplify the expression and then substitute $x = -7$ into the simplified function.

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Q5. Find $\lim_{x \to 2} x$.

Background

Topic: Basic Limits

This question tests your understanding of the limit of a linear function as $x$ approaches a specific value.

Key Terms and Formulas:

• Limit of a linear function: For $f(x) = x$, $\lim_{x \to a} x = a$.

Step-by-Step Guidance

1. Recognize that $f(x) = x$ is continuous everywhere.

2. For continuous functions, the limit as $x$ approaches a value is simply the function evaluated at that value.

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Q6. Find $\lim_{x \to 0} (4x^2 - x^3 + 7x)$.

Background

Topic: Limits of Polynomial Functions

This question tests your ability to evaluate limits for polynomial functions, which are continuous everywhere.

Key Terms and Formulas:

• Limit of a polynomial: For any polynomial $p(x)$, $\lim_{x \to a} p(x) = p(a)$.

Step-by-Step Guidance

1. Identify the function as a polynomial.

2. Substitute $x = 0$ directly into the function to evaluate the limit.

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Q7. Find $\lim_{x \to -7} \frac{x^2 + 7x}{x^2 - 49}$.

Background

Topic: Limits Involving Indeterminate Forms

This question is similar to Q4 and tests your ability to handle $\frac{0}{0}$ forms by factoring and simplifying.

Key Terms and Formulas:

• Indeterminate form: $\frac{0}{0}$

• Factoring: $x^2 - 49 = (x + 7)(x - 7)$

Step-by-Step Guidance

1. Substitute $x = -7$ to check for indeterminate form.

2. Factor numerator and denominator, cancel common factors, and then substitute $x = -7$ into the simplified expression.

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Q8. Find $\lim_{x \to 3} (x + 3)$.

Background

Topic: Limits of Linear Functions

This question tests your understanding of limits for simple linear functions.

Key Terms and Formulas:

• Limit of a linear function: $\lim_{x \to a} (x + b) = a + b$.

Step-by-Step Guidance

1. Recognize that $x + 3$ is continuous everywhere.

2. Substitute $x = 3$ directly into the function to evaluate the limit.

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Q9. Find $\lim_{x \to 2} \frac{x^2 + 7x - 8}{x^2 - 4}$.

Background

Topic: Limits Involving Rational Functions

This question tests your ability to evaluate limits for rational functions, possibly involving factoring and simplifying if you encounter an indeterminate form.

Key Terms and Formulas:

• Indeterminate form: $\frac{0}{0}$

• Factoring: $x^2 - 4 = (x - 2)(x + 2)$

Step-by-Step Guidance

1. Substitute $x = 2$ to check for indeterminate form.

2. If you get $\frac{0}{0}$, factor numerator and denominator, cancel common factors, and substitute $x = 2$ into the simplified expression.

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Q10. Find $\lim_{x \to -1} \frac{x^2 + x}{x + 1}$.

Background

Topic: Limits Involving Rational Functions

This question tests your ability to evaluate limits for rational functions, especially when direct substitution leads to an indeterminate form.

Key Terms and Formulas:

• Indeterminate form: $\frac{0}{0}$

• Factoring: $x^2 + x = x(x + 1)$

Step-by-Step Guidance

1. Substitute $x = -1$ to check for indeterminate form.

2. If you get $\frac{0}{0}$, factor numerator and denominator, cancel common factors, and substitute $x = -1$ into the simplified expression.

Try solving on your own before revealing the answer!