Business Calculus Course Review – Step-by-Step Study Guidance

Q1. Find the limit: $\lim_{x \to -5} (x^2 - 25x + 5)$

Background

Topic: Limits and Continuity

This question tests your understanding of how to evaluate limits of polynomial functions as $x$ approaches a specific value.

Key Terms and Formulas

• Limit: The value that a function approaches as the input approaches some value.

• Polynomial: A function involving only non-negative integer powers of $x$.

Step-by-Step Guidance

1. Recognize that for polynomials, the limit as $x$ approaches a value is simply the function evaluated at that value (if the function is defined there).

2. Substitute $x = -5$ into the expression $x^2 - 25x + 5$.

3. Calculate $(-5)^2$, $-25 \times (-5)$, and add 5.

Try solving on your own before revealing the answer!

Q2. Find the limit: $\lim_{x \to -3} (6x + 3)$

Background

Topic: Limits of Linear Functions

This question checks your ability to evaluate the limit of a linear function as $x$ approaches a given value.

Key Terms and Formulas

• Linear Function: A function of the form $ax + b$.

• Limit Law: For continuous functions, $\lim_{x \to a} f(x) = f(a)$.

Step-by-Step Guidance

1. Identify the function: $6x + 3$.

2. Substitute $x = -3$ into the function.

3. Calculate $6 \times (-3)$ and add 3.

Try solving on your own before revealing the answer!

Q3. Find the limit: $\lim_{h \to 0} (3x + h - 2 - 3x - 2h)$

Background

Topic: Limits and Simplification

This question tests your ability to simplify an expression before evaluating the limit as $h$ approaches 0.

Key Terms and Formulas

• Limit: The value a function approaches as the input approaches a value.

• Simplification: Combine like terms before substituting the limit value.

Step-by-Step Guidance

1. Expand and simplify the expression inside the limit: $3x + h - 2 - 3x - 2h$.

2. Combine like terms to reduce the expression to terms involving $h$ only.

3. Substitute $h = 0$ into the simplified expression.

Try solving on your own before revealing the answer!

Q4. Find the limit: $\lim_{x \to 1} (3x^3 - 4x^2 + 2x + 3)$

Background

Topic: Limits of Polynomial Functions

This question tests your ability to evaluate the limit of a polynomial function as $x$ approaches a specific value.

Key Terms and Formulas

• Polynomial: A function involving powers of $x$.

• Limit Law: For polynomials, $\lim_{x \to a} f(x) = f(a)$.

Step-by-Step Guidance

1. Identify the function: $3x^3 - 4x^2 + 2x + 3$.

2. Substitute $x = 1$ into the function.

3. Calculate each term: $3 \times 1^3$, $-4 \times 1^2$, $2 \times 1$, and add 3.

Try solving on your own before revealing the answer!

Q5. Find the limit: $\lim_{h \to 0} x + h^2$

Background

Topic: Limits Involving Powers

This question tests your understanding of how to evaluate limits involving powers as the variable approaches zero.

Key Terms and Formulas

• Limit: The value a function approaches as the input approaches a value.

• Power: $h^2$ means $h$ squared.

Step-by-Step Guidance

1. Recognize that as $h \to 0$, $h^2 \to 0$.

2. Substitute $h = 0$ into the expression $x + h^2$.

3. Simplify the result to express in terms of $x$ only.

Try solving on your own before revealing the answer!