Limits and Derivatives Formulas: Properties & Applications

Content Curator

Limits and Derivatives are two major parts of differentiation and calculus. A limit can be defined as a value that a function is seen to approach as the input, yielding some value in return. The rate at which a function or quantity changes in relation to others can be termed as its derivative.

A limit is a value that a function approaches as the input (otherwise also known as, an independent variable) approaches a certain value.
A derivative measures the rate at which a function changes. It can be represented as the slope of the tangent line to a curve at a specific point.
Limits and derivatives are very closely related to one another.
The derivative of a function can be indicated as the limit of the difference quotient as the change in x approaches zero.

Table of Content

What is Limit?
Limits Formula
What is Derivative?
Derivatives formula
Application of Limits and Derivatives
Solved Examples
Things to Remember
Previous Year's Questions
Sample Questions

Key Terms: Limits, Derivatives, Calculus, Function, Continuity, Constant, Integration, Variable, Definite Integral

What is Limit?

[Click Here for Sample Questions]

When a function's limit, f(x), reaches a certain value, the function is said to have reached its limit. Integral calculus, integration, and function continuity are all defined in terms of limits.

Limit represents the behaviour of a function or sequence near a particular input or index value, without necessarily being equal to the function value at that point.
In the event where f(y) is a function, the limit of the function is denoted by, lim_y→c. Here,c can be any constant value.

To define a limit, the following notation is used:

lim_{x → a} f(x) = L

This notation represents the limit of the function f(x) as x approaches the value a.
The value L is the limit of the function as x approaches “a” if, for any positive number ε, there exists a corresponding positive number δ such that if 0 < |x-a| < δ, then |f(x)-L| < ε.

Limits Formula

[Click Here for Previous Year's Questions]

We represent a function's limit as follows to clarify it:

lim_x→af(x)

Left Hand Limit

The unique number so obtained is known as the f(x) left-hand limit at x = a; we express it as x = a if the function values at the point extremely near to a, on the left, tend to a definite unique number as x goes to a.

f(a − 0) = lim_x→a−f(x) = lim_h→0f(a − h)

Right Hand Limit

f(a + 0) = lim_x→a+f(x) = lim_h→0f(a + h)

Existence of Limits

lim_x→af(x)lim_x→af(x) exists, if

(i) lim_x→a−f(x)lim_x→a−f(x) and lim_x→a+f(x)lim_x→a+f(x) both exists

(ii) lim_x→a−f(x) = lim_x→a+f(x)

Properties of Limits

The properties of limits can be described by the given formulas:

lim_x→a[p(x) + g(x)] = lim_x→ap(x) + lim_x→ag(x)
lim_x→a[p(x) − g(x)] = lim_x→ap(x) − lim_x→ag(x)
For every real number K, lim_x→a[kp(x)] = klim_x→ap(x)
lim_x→a[p(x)q(x)] = lim_x→ap(x) × lim_x→aq(x)
lim_x→ap(x) / q(x) = lim_x→ap(x) / lim_x→aq(x)

Limits of trigonometric functions:

If p and q are real-valued functions with the same domain, such that, p(x) ≤ q(x) for all the values of x. For a value b, if both lim_x→a p(x) and lim_x→a q(x) exists then,

lim_x→a p(x) ≤ lim_x→a q(x)

General Formulas for Derivatives

The general formulas for derivatives can be shown as:

lim _x->0 sin x = 0
lim _x->0 cos x = 0
lim _x->0 sin x / x= 0
lim _x->0 tan x / x = 1
lim _x->0 1-cos x / x = 0
lim _x->0 sin^-1 x / x = 1
lim _x->0 tan^-1 x / x = 1
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, -∞

What is Derivative?

[Click Here for Sample Questions]

The rate at which a function or quantity changes in relation to others is referred to as its derivative.

In calculus, the derivative is a concept that measures how much a function changes as its input variable changes.
It is the slope of a tangent line at any point on a curve or function.
The derivative of a function f(x) is denoted by f'(x) or dy/dx, and it represents the instantaneous rate of change of the function at a given point.

The derivative formula can be expressed as follows:

lim_a→0 \(\frac{f(x + a) − f(x)}{a}\)

A function's derivative is represented by the symbol f' (x). Let's now examine the characteristics of derivatives.

Limits and Derivatives Detailed Video Explanation

Derivatives Formula

[Click Here for Previous Year's Questions]

If f is a real-valued function, as assumed, then

f′(x) = lim_x→a f(x + h) – f(x) / h is referred to as the f at x if derivative lim_x→a f(x + h) - f(x) / h exists on a finite scale.

For function f, its derivative is defined as f′(x), assuming the existence of the aforementioned equation. Look up all the derivative formulas for trigonometric, inverse, hyperbolic, and other functions here.

Properties of Derivatives

The properties of derivatives can be shown in the form of formulas:

\(\frac{d}{dx}\) [p(x) + q(x)] = \(\frac{d}{dx}\) [p(x)] + \(\frac{d}{dx}\) [q(x)]
\(\frac{d}{dx}\) [p(x) − q(x)] = \(\frac{d}{dx}\) [p(x)] – \(\frac{d}{dx}\) [q(x)]
\(\frac{d}{dx}\) [p(x) × q(x)] =\(\frac{d}{dx}\) [p(x)]q(x) + p(x) \(\frac{d}{dx}\) [q(x)]
\(\frac{d}{dx}\) [p(x) / q(x)] = {\(\frac{d}{dx}\) [p(x)]q(x) − p(x) \(\frac{d}{dx}\) [q(x)] / (q(x))²

General Formulas for Derivatives

The general formulas for derivatives are:

\(\frac{d}{dx}\) (xⁿ) = nxⁿ⁻¹
\(\frac{d}{dx}\) (sin x) = cos x
\(\frac{d}{dx}\) (cos x) = −sin x
\(\frac{d}{dx}\) (tan x) = sec² x
\(\frac{d}{dx}\) (cot x) = −cosec² x
\(\frac{d}{dx}\) (sec x) = sec x tan x
\(\frac{d}{dx}\) (cosec x) = −cosec x cot x
\(\frac{d}{dx}\) (a^x) = a^xlog^ea
\(\frac{d}{dx}\) (e^x) = e^x
\(\frac{d}{dx}\) (log^ex) = 1 / x

Application of Limits and Derivatives

[Click Here for Sample Questions]

The following are some of the most significant uses of class 11 limit and derivative formulas:

Business models use limit and derivative class 11 formulas to compute and graphically display the profit or loss.
These formulas are essential for analyzing temperature fluctuations.
To measure the speed or distance travelled, such as in miles per hour, kilometer per hour, etc., limits and derivatives class 11 formulas are frequently used.
Formulas from Limit and Derivatives class 11 can be used to derive a number of expressions and equations in fields like physics.
Calculations based on limits and derivatives are used in seismology to determine the magnitude range of an earthquake.

List of All Limits and Derivatives Formulas

The list of limits and derivative class 11 formulas:

\(\frac{d}{dx}\) [f(x) + g (x)] = \(\frac{d}{dx}\) [f(x)] + \(\frac{d}{dx}\) [g(x)]
\(\frac{d}{dx}\) [f(x) - g (x)] = \(\frac{d}{dx}\)[f(x)] - \(\frac{d}{dx}\) [g(x)]
\(\frac{d}{dx}\) [f(x) * g (x)] = \(\frac{d}{dx}\) [f(x)]*[g(x)] + [f(x)]*\(\frac{d}{dx}\) [g(x)]
\(\frac{d}{dx}\)[f(x) / g (x)] = {\(\frac{d}{dx}\) [f(x)]*[g(x)] - [f(x)]*\(\frac{d}{dx}\) [g(x)]} / g(x)²
lim_{x → a}\(\frac{(x^n - a^n)}{x - a}\) = na^n-1
lim_{x → a}\(\frac{Sin x}{x}\) = 1
lim_{x → a} \(\frac{1 - cos x}{x}\) = 0
f ´( x ) = \(\frac{d f(x)}{dx}\)= lim_{h → 0} \(\frac{f(x+h) - f(x) }{h}\)

Solved Examples

[Click Here for Previous Year Questions]

Ques. Evaluate the following: d/dx(x²+ 4).

Ans. We are aware that, \(\frac{d}{dx}\)(xn) = n x^n-1 and \(\frac{d}{dx}\)(c) = 0.

Thus, \(\frac{d}{dx}\)(x² + 4)= 2x

Ques. Determine the value of lim_x→0x²+ 1.

Ans. We have considered: lim_x→0x²+ 1

Now, we put x= 0 directly, thus we acquire the value of limit as 1.

Previous Year Questions

Which of the following is divisible by … [AP EAPCET]
If x>0 … [BITSAT 2018]
If the function f(x) defined by … [KCET 2014]
If f(x) … [KEAM]
Let p … [JEE Main 2016]
Which the volume increases (in cubic centimeters per minute) when the radius is 5 centimetres is …
The equation of the tangent to the conic x2−y2−8x+2y+11=0 at (2,1) is …
Angle between y2=x and x2=y at the origin is …

Things to Remember

When a function's limit, f(x), reaches a certain value, the function is said to have reached its limit.
We represent a function's limit as follows to clarify it: lim_x→af(x)
The rate at which a function or quantity changes in relation to others is referred to as its derivative.
The derivative formula can be expressed as follows: lim_a→0 f(x + a) − f(x) / a
A function's derivative is represented by the symbol f' (x).
Business models use limit and derivative class 11 formulas to compute and graphically display the profit or loss.

Read also:

Topic Related Links
Calculus	Limit formula	Application of derivatives
Limits	Continuity and Differentiability	Limits and continuity

Sample Questions

Ques. How do we solve limits? (2 marks)

Ans. The multiple Limits of a Function are as follows:

Utilizing direct substitution, evaluate limits
Utilize factoring and cancelling to evaluate limits.
Expand and simplify to evaluate limits
Use fractional combinations to evaluate limits.
Limits should be assessed by multiplying by the conjugate.

Ques. How do limits translate into mathematical terms? (2 marks)

Ans. The limit of a function f(x), as x approaches a, is identical to L, as written in mathematics. If we take x to be sufficiently close to "a" (on either side of a) but not equivalent to a, we can make the values of the function f(x) arbitrarily close to L. This means that the value of f(x) approaches the number 'L' considerably more closely when the value of 'x' approaches the number a (from either side of a). Keep in mind that we never consider x = a while calculating the limit of f(x) as x approaches. When x = a, it is not even necessary to define F(x).

Ques. What are the Practical Applications of Limits and Derivatives Formulas? (1 mark)

Ans. In real life, formulas for limits and derivatives are used to estimate a variety of quantities. Engineers, for instance, will approximate a function using minor variations in the function and then calculate the derivative of the function by increasing the spacing between the function sample intervals.

Ques. What Do Calculus Limits Mean? (1 mark)

Ans. The value that a function approaches as its inputs become closer and closer to a certain number is described by a limit. The fundamental concept behind all calculus differentials and integrals is the concept of a limit.

Ques. Which expression's limit is the following: lim_x→5 x²−5 / x²+x−30? (2 marks)

Ans. The stated limit is the product of two polynomials, where x = 5. This unquestionably equalizes the denominator and the numerator to zero (0). As illustrated below, we must factor both the numerator and the denominator.

lim_x→5 (x−5)(x+5) / (x−5)(x+6)

Simplify the phrase to obtain:

lim_x→5 x+5 / x+6

= 10 / 11

Ques. Evaluate the function: (x² – 9) / x – 3 (2 marks)

Ans. The function's limit in this case lies in the number 00.

The result is what we obtain when we represent the numerator as the product of two terms.

= (x + 3)(x – 3) / x – 3

= 3 + 3

= 6.

Ques. Using the first principle, determine the derivative of f(x) = x³. (3 marks)

Ans. According to the first principle,

f'x = f(x+h) − f(x) / h

When we enter f(x) = x³ into the equation above, we obtain,

f'x = (x+h)³ − (x)³ / h

f'x = x³+ h³+ 3xh (x+h)−x³ / h

f'x = h²+ 3x(x+h)

Substituting h = 0 we get,

f’x = 3x²

Ques. Solve lim_x→2 (sin 2x/x). (2 marks)

Ans. Given, lim_x→2 (sin 2x/x)

We can write it as;

lim_x→2 (sin 2x/2x) × 2

Since, lim_x→2 (sin x/x) = 1

Therefore, lim_x→2 (sin 2x/2x) × 2 = 1 × 2 = 2

Ques. Find the positive integer “n” so that lim_{x → 3}[(xⁿ– 3ⁿ)/(x – 3)] = 108. (3 marks)

Ans. The limit that is given is, lim_{x → 3}[(xⁿ– 3ⁿ)/(x – 3)] = 108

Here, we have:

lim_{x → 3}[(xⁿ– 3ⁿ)/(x-3)] = n(3)^n-1

→ n(3)^n-1= 108

It can be expressed as:

n(3)^n-1= 4 (27) = 4(3)^4-1

Hence, by comparing the exponents in the above equation, we can obtain:

n = 4

Consequently, the value of the positive integer “n” can be seen as 4.

Ques. Determine the derivative of cosx/(1+sin x). (5 marks)

Ans. As per the given question, cosx/(1+sin x)

Now, let y = cosx/(1+sin x)

As per the question, we need to differentiate the function with respect to “x”, we obtain:

\(\frac{dy}{dx}\) = \(\frac{d}{dx}\) (cos x/(1+sin x))

Now, use the u/v formula in the above form, we get

⇒ \(\frac{dy}{dx}\)= [(1+sin x)(-sin x) – (cos x)(cos x)]/(1+sin x)²

⇒ \(\frac{dy}{dx}\) = (-sin x – sin²x-cos²x)/(1+sin x)²

Now, take (-) outside from the numerator, we get:

⇒ \(\frac{dy}{dx}\) = -(sin x + sin^2.x + cos²x)/(1+sin x)²

We know that sin^2.x + cos²x = 1

Now, by replacing the same, we get:

⇒ \(\frac{dy}{dx}\) = -(1+sin x)/(1+sin x)²

Now, cancel (1+sin x) from the numerator and denominator, thus:

⇒ \(\frac{dy}{dx}\) = -1/(1+sin x)

Hence, the derivative of cosx/(1+sin x) is -1/(1+sin x).

For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates

Read also:

Class 11 and 12 Study Guides
NCERT Solutions Class 6 to 12	Class 11 Notes	NCERT Solutions for Class 11 English
Maths MCQs	NCERT Solutions for Class 12 Physics	Comparison Topics in Maths
Class 11 PCMB Syllabus	Topics with relation in physics	Physics Study Notes
NCERT Class 12 Biology Book	Physics MCQs	Important Physics Constants and Units
Class 12 Biology Notes	NCERT Solutions for Class 11 Physics	NCERT Solutions for Class 12 Biology
Biology Study Notes	NCERT Solutions for Class 12 Math	Class 12 Chemistry Practicals
SI units in Physics	Class 12 Chemistry Notes	MCQ questions in physics
NCERT Solutions for Class 11 Biology	NCERT Solutions for Class 12 English	NCERT Class 11 Chemistry Book
Class 12 Maths Notes	Class 12 PCMB Notes	NCERT Class 12 Maths Book
NCERT Class 12 Textbooks	NCERT Solutions for Class 11 Math	Comparison Topics in Biology
NCERT Solutions for Class 11 Chemistry	Important Chemistry Formulas	NCERT Solutions for Class 12 Chemistry
Biology MCQs	NCERT Class 11 Physics Book	Class 12 Physics Practicals
NCERT Class 11 Biology Book	Comparison in Physics	Class 12 Physics Book PDF
Math Formulas	Chemistry Study Notes	Important Derivations in Physics
Class 12 Physics Notes	Important Named Reactions	Chemistry MCQs
Formulas in Physics	Important Chemical Reactions	Comparison topics in Chemistry
Periodic Table in Chemistry	Class 12 Syllabus	Class 11 Syllabus

Limits and Derivatives Formulas: Properties & Applications

What is Limit?