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Limits and Derivatives concepts offer an introduction to Calculus. The value that approaches as the input gets closer to some particular number can be called the Limit of a function.
- A changing rate of change in relation to an independent variable in any function is called Derivatives.
- It is equivalent to the slope of the line to the graph of the function.
- The limit of a function is generalized in the concept of the limit of a topological net.
- The topic is covered in detail in NCERT Class 11 Mathematics.
- Derivatives are used when varying quantities are used and the rate of change is not constant.
- Modern Physics is based on the concept of limits and derivatives.
- It is used to calculate the speed and distance at which your car is travelling.
- The Limits and derivatives of a function can be represented by:
limx→a(f(x) = L
f′(x)=limh→0(f(x+h)−f(x))/h
Key Terms: Limits and Derivatives, Limits, Derivations, Theorem of Derivatives, Derivative Functions, Theorem of Limits, Variables, Polynomial, Calculus, Functions, Limits Functions
Limits of a Function
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The concept of limit can be described as the value that approaches as the input gets closer to some particular number. The concept of Limit helps in defining integrals and continuity.
- The limits of a function describe how a function acts near a point instead of that particular point.
- It means that they do not depend upon the actual value of a function.
- The function is used to explain integrals, derivatives, and continuity.
- It is used to check values that are smaller than a limit.
Example of Limits of a FunctionExample 1: The limits of a function is used to measure the temperature of ice dipped in a warm glass of water. Example 2: To calculate the limit of: \(\displaystyle \lim _{x \rightarrow 6}\left[\frac{(x-3)(x-2)}{x-4}\right]\) \(\displaystyle \begin{aligned} &\lim _{x \rightarrow 6}\left[\frac{(x-3)(x-2)}{x-4}\right] \\ &=\left[\frac{\displaystyle \lim _{x \rightarrow 6}(x-3)\lim _{x \rightarrow 6}(x-2)}{\displaystyle \lim _{x \rightarrow 6}(x-4)}\right] \\ &=\left[\frac{(6-3)(6-2)}{(6-4)}\right] \\ &=\left[\frac{(3)(4)}{(2)}\right]=6 \end{aligned}\) |
Concept of Limit Through a Graph
Read More:
| Chapter Related Concepts | ||
|---|---|---|
| Value of e | Differentiation Rules | Exponential growth formula |
| Approximation | Application of Derivatives | Infinity |
Theorems of Limits
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The important theorems of limits are as follows:
- If f is a polynomial or a rational function, and a is in the domain of f, then
\(\displaystyle \lim_{x \rightarrow a} f(x) = f(a)\)
- If f(x) = g(x) whenever x≠a, then
\(\displaystyle \lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} g(x)\)
Limits Representation
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The limit of a function can be represented by:
\(\displaystyle \lim_{x \rightarrow a} f(x) = L\)
Properties of Limits
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Here are some properties of limits:
- \(\displaystyle \lim _{x \rightarrow 0} \frac{\sin x}{x}=\lim _{x \rightarrow 0} \frac{\tan x}{x}=1\)
- \(\displaystyle \lim _{x \rightarrow 0} \frac{\sin ^{-1}}{x}=\lim _{x \rightarrow 0} \frac{\tan ^{-1}}{x}=1\)
- \(\displaystyle \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}=1 \)
- \(\displaystyle \lim _{x \rightarrow 0} \frac{a^{x}-1}{x}=\ln _{a}\)
- \(\displaystyle \lim _{x \rightarrow 0} \frac{e^{x}-1}{x}=1\)
- \(\displaystyle \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{(x-a)}=n \cdot a^{n-1}\)
- \(\displaystyle \lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e\)
- \(\displaystyle \lim _{x \rightarrow 0} \frac{(1+x)^{m}-1}{x}=m\)
Limit Formulas
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Some basic Limit Formulas are tabulated below.
- lim x->0 sin x = 0
- lim x->0 cos x = 0
- \(\begin{array}{l} \lim_{x\to 0}\frac{\sin x}{x}=1\end{array}\)
- \(\begin{array}{l} \lim_{x\to 0}\frac{\tan x}{x}=1\end{array}\)
- lim x->0 1-cos x/x = 0
- \(\begin{array}{l}\lim_{x\to 0}\frac{\sin^{-1}x}{x}=1\end{array}\)
- \(\begin{array}{l} \lim_{x\to 0}\frac{\tan^{-1}x}{x}=1\end{array}\)
- lim x->a sin-1 x = sin-1 a,|a|≤1
- lim x->a cos-1 x = cos-1 a,|a|≤1
- lim x->a tan-1 x = tan-1 a, -∞
Limits of form 1\(\infty\)
The limits of Form 1\(\infty\) are:
- \(\begin{array}{l} \lim_{x\to 0}(1+x)^{\frac{1}{x}}=e\end{array}\)
- \(\begin{array}{l} \lim_{x\to \infty }(1+\frac{1}{x})^{x}=e\end{array}\)
- \(\begin{array}{l}\lim_{x\to \infty }(1+\frac{a}{x})^{x}=e^{a}\end{array}\)
Derivatives of a Function
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Derivatives of a function can be described as a changing rate of change with relation to an independent variable in any function. This concept can be used in various practical situations, like- predicting future market values or any particular market positions.
- Derivatives is also known as differential coefficient of y with respect to x.
- It is used to optimize (maximize/minimize) a function.
- Derivatives are used to determine when the intervals of the function are increasing or decreasing.
- It can also be calculated by using differentiation by the first principle.
Examples of Derivatives of a FunctionExample 1: The function of derivative is used to measure the magnitude of earthquake. Example 2: If we need to find the derivative of the sin x at x = 0, Suppose, f(x) = sin x, then, f'(0) = \(\displaystyle \lim_{h \rightarrow 0} \frac{f(0+h) - f(0)}{h}\) = \(\displaystyle \lim_{h \rightarrow 0} \frac{\sin (0+h) - \sin(0)}{h}\) = \(\displaystyle \lim_{h \rightarrow 0} \frac{\sin h}{h}\) = 1 |
Theorem of Derivatives
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The important theorem of derivatives are as follows:
- The derivative of the sum of two functions is the sum of the derivatives of the functions.
d/dx[f(x)+g(x)]=d/dxf(x)+d/dxg(x)
- The derivative of the difference between two functions is the difference between the derivatives of the functions.
d/dx [f(x)–g(x)]=d/dx f(x)–d/dx g(x)
- The derivative of the product of two functions is given by the Product Rule.
d/dx [f(x).g(x)]=[d/dx f(x)].g(x)+[d/dx g(x)].f(x)
Derivatives Representation
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The basic derivatives representation is as follows:
\(\displaystyle f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)
Properties of Derivatives
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There are some properties of derivatives-
- The limit of a sum is the sum of the limits:
Lim x→a (f(x)±g(x)) = lim x→a f(x) ± lim x→a g(x).lim x→a (f(x)±g(x)) = lim x→a f(x)±lim x→a g(x)
- The limit of a product is the product of the limits:
Lim x→a f(x)g(x) = (lim x→a f(x)) (lim x→a g(x)).Lim x→a f(x)g(x) = (lim x→af(x))(lim x→a g(x))
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Derivatives Formulas
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The important derivatives formula are as follows:
Derivative Formulas for Elementary Functions
Some important formulas for elementary function are as follow:
- d/dx (k) = 0, where k is any constant
- d/dx(x) = 1
- d/dx(xn) = nxn-1
- d/dx (kx) = k, where k is any constant
- d/dx (√x) = 1/2√x
- d/dx (1/x) = -1/x2
- d/dx (log x) = 1/x, x > 0
- d/dx (ex) = ex
- d/dx (ax) = ax log a
Derivative Formulas for Trigonometric Functions
We may also compute the derivative of trigonometric functions, such as sin, cos, and tan. The formulae are as follows:
- d/dx (sin x) = cos x
- d/dx (cos x) = -sin x
- d/dx (tan x) = sec2x
- d/dx (cosec x) = -cosec x cot x
- d/dx (sec x) = sec x tan x
- d/dx (cot x) = -cosec2
Things to Remember
- The concepts of Limits and Derivations Formulas are based on the ideas of Sir Issac Newton’s.
- Limit is the value that approaches as the input gets closer to some particular number and can be represented as \(\displaystyle \lim_{x \rightarrow a} f(x) = L\)
- Derivatives is the limit of the average rate of change in the function as the length of the interval on which the average is computed tends to zero.
- NCERT Solutions For Class 11 Maths Chapter 13: Limits and Derivatives will help you solve complex problems easily.
- For preparation, individuals can try out Important Questions for Class 11 Maths Chapter 13: Limits and Derivatives
Read More:
| Class 11 Mathematics Related Concepts | ||
|---|---|---|
| Increasing and Decreasing Functions | Logarithmic Differentiation | Maxima and Minima |
| Implicit Function Differentiation | Application of Integrals | Quotient Rule |
Sample Questions
Ques. Evaluate the following limit: (2 marks)
\(\displaystyle \lim_{x \rightarrow 3} (x+3)\)
Ans:
\(\displaystyle \begin{align*} \lim_{x \rightarrow 3} (x+3) &= (3 + 3) \\ &= 6 \end{align*}\)
Ques. Compute the derivative of (2 marks)
f(x) = sin2 x.
Ans. (df(x))/(d(x)) = d/dx{sin x(sin x)}
= (sin x)’sin x + sin x(sin x)’
= (cos x)sin x + sin x(cos x)
= 2 sin x cos x = sin 2x
Ques. Compute the derivative of (2 Marks)
f(x) = (x+1)/x
Ans. Let u = x+1 and v = x
df(x)/dx = d/dx (x+1/x) = dx/d (u/v)
= (1(x) – (x+1)1)/x2
= -1/x2
Ques. Explain the concept of Limits. (2 Marks)
Ans. The concept of limits can be defined as- the value that approaches as the input gets closer to some particular number. We can show this concept through the following graph:
Ques. How can you define Derivatives? What is the Derivatives Formula? (3 Marks)
Ans.Derivatives can be defined as- “the limit of the average rate of change in the function as the length of the interval on which the average is computed tends to zero.”
Derivatives Formula: \(\displaystyle f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)
Ques. What is the difference between derivative and integral? (2 Marks)
Ans. The function defines the slope of the curve is called Derivatives, whereas the area under the curve of the function is called an integral of a function defines.
Ques. Evaluate the function \(\displaystyle \lim_{x \rightarrow 3} \frac{x^2-9}{x-3}\). (3 marks)
Ans. We know, the limit of the given function is \(\frac{0}{0}\).
Now, by representing the numerator as the product of two terms, we get
\(\displaystyle \lim_{x \rightarrow 3} \frac{x^2-9}{x-3} = \lim_{x \rightarrow 3} \frac{(x+3)(x-3)}{x-3}\)
= \(\displaystyle \lim_{x \rightarrow 3} (x+3)\)
= 3 + 3
= 6
Ques. Find the derivative of the function f(x) = 5x2 – 2x + 6. (4 Marks)
Ans. Given, f(x) = 6x2 – 2x + 6
Now taking the derivative of f(x),
d/dx f(x) = d/dx (6x2 – 2x + 6)
Let us split the terms of the functi?on is:
d/dx f(x) = d/dx (6x2) – d/dx (2x) + d/dx (6)
Using the formulas:
d/dx (kx) = k and d/dx (xn) = nxn – 1
⇒ d/dx f(x) = 6(2x) – 2(1) + 0 = 12x – 2
Ques. Compute the derivative of (2 Marks)
f(x) = (x+12)/x
Ans. Let u = x+1 and v = x
df(x)/dx = d/dx (x+12/x) = dx/d (u/v)
= (1(x) – (x+12)1)/x2
= -12/x2
Ques. Find the derivative of 2 cos x + 1 (2 Marks)
Ans. Let the given function be f(x) = 2 cos x + 1
Now, taking the derivative,
d/dx f(x) = d/dx (2 cos x + 1)
= d/dx (2 cos x) + d/dx (1)
= 2 (-sin x) + 0
= – 2 sin x
Ques. Evaluate lim x->3 (3x3 – 2x2 +2) (2 Marks)
Ans. This can written in the form of ,
=lim x->3 (3x3)- lim x->3 (2x2)+ lim x->3 (2)
= 2lim x->3 (x3)- 3lim x->3 (x2)+ (2)
= 3(33)- 2(32)+2
= 3*27- 2*9+1
= 64
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