What is the limit of (x² - 4)/(x - 2) as x approaches 2?

The limit is 4. Factor x² - 4 as (x - 2)(x + 2), cancel x - 2, then evaluate x + 2 at 2.

How do you differentiate x² sin(x)?

Use the product rule: d/dx[x²sin(x)] = 2xsin(x) + x²cos(x).

What is the limit of (x - 1)/(x² - 1) as x approaches 1?

The limit is 1/2. Factor x² - 1 as (x - 1)(x + 1), cancel x - 1, then substitute 1.

Free Course Image Calculus 1 for beginners

Free online course Calculus 1 for beginners

Duration of the online course: 10 hours and 40 minutes

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Build confidence in calculus with this free online course: master limits and derivatives through clear explanations and practice questions, and earn a certificate.

In this free course, learn about

Core Calculus I topics: functions, limits, derivatives, and basic applications
Evaluate limits as x→∞ for polynomials using dominant-term behavior
Compute limits near a point via algebraic simplification and factoring
Handle removable discontinuities like (x^2-4)/(x-2) by canceling factors
Compute limits of rational expressions such as (x-1)/(x^2-1) as x→1
Differentiate products like f(x)=x^2·sin(x) using the product rule
Know conditions for lim f(x)/g(x) as x→a (e.g., g(x)≠0 near a, limit laws)

About the free online course

Calculus can feel intimidating at first, especially when symbols, functions, and unfamiliar rules appear all at once. This free online course is designed to help you start calmly and build real understanding step by step, even if you are just beginning. You will learn how to think like a calculus student: focus on what a function is doing, what happens near a point, and how to describe change with simple, reliable methods.

The course puts special emphasis on limits, because limits are the foundation behind many ideas in Calculus 1. You will practice evaluating limits as x approaches specific values and as x grows without bound, developing an intuition for end behavior and local behavior. Along the way, you will see how algebraic simplification resolves expressions that look undefined at first glance, and you will learn to interpret common limit forms with confidence.

Once that foundation is in place, you will move into derivatives and discover how they describe instantaneous rate of change. The course guides you through core derivative skills, including working with products and combining algebra with trigonometric functions. Rather than memorizing rules in isolation, you will practice choosing an approach, carrying out the steps carefully, and checking whether your result makes sense.

Practice is built into the experience through targeted exercises that mirror typical school and exam questions. This helps you turn passive watching into active problem solving, so you can spot patterns, avoid common mistakes, and improve speed without sacrificing understanding. If you are preparing for school assessments, returning to math after a break, or aiming to strengthen prerequisites for science, engineering, or economics, this course provides a clear, supportive path into Calculus 1.

By the end, you should feel more comfortable reading calculus notation, evaluating limits, and differentiating functions, with stronger intuition about what the mathematics is saying. Learn at your own pace, revisit tricky ideas as needed, and build a solid starting point for more advanced topics.

Course content

Video class: Calculus 1 - full course for beginners 10h40m
Exercise: What is the limit of the function f(x) = 2x^3 - 3x^2 + x + 7 as x approaches infinity?
Exercise: What is the derivative of the function f(x) = x^2 * sin(x)?
Exercise: Consider the following function: f(x) = (x^2 - 4)/(x - 2). What is the limit as x approaches 2?
Exercise: What is the limit of the function f(x) as x approaches 1 for f(x) = (x - 1)/(x^2 - 1)?
Exercise: What is the limit of the following function as x approaches 2: (x^2 - 4) / (x - 2)?
Exercise: Consider the expression for \\( \lim_{{x \to a}} \frac{{f(x)}}{{g(x)}} \\). Which of the following statements about the limit is true?