Limits and Derivatives
A car shows 60 km/h on the dial, yet it never drives for a single instant.
- How can a speed exist at one moment?
- The answer is a limit.
- This chapter starts with the idea of a limit, learns to find limits by algebra and by the standard limits, and then uses a limit to define the derivative: the slope of a tangent and the rate of change at an instant.
- You differentiate polynomials and trigonometric functions from first principles and with the product and quotient rules.
- The last course takes the harder limits that exams love, with the number e, the squeeze theorem, series expansions and L'Hôpital's rule.
- It covers the unit of the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Part of Mathematics · High School (11–12) (chapter 20)
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Limits and Standard Limits
How close can you get to a place you may not stand on? A limit is the value a function heads for as x comes close to a point, even when the function has a hole at that point. You read limits from a table and a graph, from the left and the right, and you see when a limit does not exist. Then you find them by factorising, rationalising and dividing by the highest power, and you learn the standard limits of sine, powers, exponentials and logarithms. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level, IB and AP.
Derivatives from First Principles to the Product Rule
A speedometer shows how fast the distance changes at one instant. A tangent shows how steep a curve is at one point. Both are the same limit. Draw a secant line, move its second point closer and closer, and the secant becomes the tangent. The slope is the derivative. You find it from the definition for powers, sine, cosine and tangent, then speed up with the sum, product and quotient rules, and use the derivative as a rate. It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.