Chapter 13
LIMITS AND DERIVATIVES
Some useful results
(a² – b² ) = (a + b) (a – b)
a³ – b³ = (a – b) (a² + ab + b² )
a³ + b³ = (a + b) (a² – ab + b² )
a4 – b4 = (a² – b² ) (a² + b² ) = (a + b) (a – b) (a² + b² )
Key Concepts
- The expected value of the function as dictated by the points to the left of a point defines the left-hand limit of the function at that point. The limit f(x) is the expected value of f at x = a given the values of f near x to the left of a.
- The expected value of the function as dictated by the points to the right of point a defines the right- hand limit of the function at that point. The limit f(x) is the expected value of f at x = a given the values of f near x to the left of a.
- Let y = f(x) be a function. Suppose that a and L are numbers such that as x gets closer and closer to a, f(x) gets closer and closer to L we say that the limit of f(x) at x = a is L, i.e., fx =L.
- Limit of a function at a point is the common value of the left- and right-hand limit if they coincide, i.e., fx=fx .
- Real life examples of LHL and RHL
- If a car starts from rest and accelerates to 60 km/hr in 8 seconds, which means the initial speed of the car is 0 and it reaches 60 km 8 seconds after the start. On recording the speed of the car, we can see that this sequence of numbers is approaching 60 km in such a way that each member of the sequence is less than 60. This sequence illustrates the concept of approaching a number from the left of that number.
- Boiled milk which is at a temperature of 100 degrees is placed on a shelf; temperature goes on dropping till it reaches room temperature.
As the time duration increases, temperature of milk, t, approaches room temperature say 30° .This sequence illustrates the concept of approaching a number from the right of that number.
8. Limit of a polynomial function can be computed using substitution or algebra of limits.
9. The following methods are used to evaluate algebraic limits:
- Direct substitution method
- Factorization method
- Rationalization method
- By using some standard limits
- Method of evaluation of algebraic limits at infinity
10. For computing the limit of a rational function when direct substitution fails, use factorisation , rationalisation or the theorem.
11. Let f and g be two real valued functions with the same domain such that f(x) ≤ g(x) for all x in the domain of definition. For some a, if both fx and gx exist, then fx ≤gx .
12. Let f, g and h be real functions such that f(x) ≤ g(x) ≤ h(x) for all x in the common domain of definition. For some real number a, if fx .= ℓ= hx then gx =l
Provided this limit exists and is finite. Derivative of f(x) at a is denoted by f’(a).
14. A function is differentiable in its domain if it is always possible to draw a unique tangent at every point on the curve.
15. Finding the derivative of a function using definition of derivative is known as the first principle of derivatives or ab-initio method.
16. Differentiation of a constant function is zero.
17. If f(x) is a differentiable function and ‘c’ is a constant, then ddxc.fx=c.ddxfx.
18. Let f and g be two functions such that their derivates are defined in a common domain. Then
i. Derivative of the sum of two functions is the sum of the derivatives of the functions.
19. Generalization of the product rule: Let f(x), g(x) and h(x) be three differentiable functions. Then
20. Derivative of f(x) = xⁿ is nxn−1 for any positive integer n.
23. Standard derivatives
24. The derivative is the instantaneous rate of change in the terms of Physics and is the slope of the tangent at a point.
25. A function is not differentiable at the points where it is not defined or at the points where the unique tangent cannot be drawn.
26. Conisider that f’(x), dydx,dfxdx and y’ are all different notations for the derivative with respect to x.
Steps for finding the left-hand limit
3. Step 3: Using appropriate formula simplify the given function.
4. Step 4: The final value is the left-hand limit of the function at x = a.
Steps for finding the right-hand limit
2. Step 2: Substitute x = a – h and replace x→a+ by h →0 to get f(a+h)
3. Step 3: Using appropriate formula simplify the given function.
4. Step 4: The final value is the left-hand limit of the function at x = a.
Steps for factorisation method
2. Step 2: Factorize f x and g x.
3. Step 3: Cancel out the common factors.
4. Step 4: Use the direct substitution method to find the final limit.
Steps for rationalisation method
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