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Limit formula, in maths, deals with the concepts of Calculus where the values at some point are determined, which may not be deterministic otherwise. Limits are the fundamental concepts in mathematics that are used to find out the particle state at a particular position, and to find the initial and final positions of a particle to determine values.
The formula of limit is:
| \(\mathop {\lim }\limits_{x \to a} f(x) = A \) |
Here,
- f(x) = function
- x = variable approaching to value ‘a’
The limit value typically has two types of values, namely, the Left-hand Limit and the Right-Hand Limit. Thus,
- Left-Hand Limit: \(\lim_{x\to a-} f(x) =\lim_{h\to 0}f(a-h) \)
- Right-Hand Limit: \(\lim_{x\to a+} f(x) =\lim_{h\to 0}f(a+h) \)
It is derived from calculus and is proven as an effective process to give outputs for given limits. Limits are also used in the analysis of the characteristics of a function near and at a particular point.
Read also: Continuity and Differentiation
| Table of Content |
Key Terms: Limit, Calculus, Independent Variable, Left-Hand Limit, Right-Hand Limit, Derivatives, Continuities, Integrals
What is a Limit?
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Limit represents the behaviour of a function at a specific point. The formula of limit helps to analyze that function. Thus, limit can be defined as:
| “The behaviour of some quantity which depends on an independent variable which approaches or tends to come close to a particular value” |
- In mathematics, limit simply gives the near values or high values of the output.
- These are essential to determine derivatives, continuities, and integrals of functions.
- It is represented as limx→a f(x) = b
- In the above representation, it states that if the limit approaches ‘a’, then the f(x) value is equal to b.
Representations vary based on the type of limits. Here are some examples of it,
- Right-hand side limits: It is represented as: lim1 +f(x) = 1
- Left-hand side limit: It is represented as: lim1 -f(x) = 1
- Infinite limits: In this f (x) value has no limit and it can extend to anywhere in the plane. It is represented as limx→∞ f(x) = 1
- One-sided infinite limits: Here, one side of f(x) is represented as infinity. These are represented as, lim1 + f(x) = ∞ or lim1 – f(x) = ∞
Let y = f(x) be a function of x. If at some point x = a, f(x) takes an indeterminate form, we can take the values of the function that is close to a. If these values tend to a unique number like x tends to a, then the obtained unique number is known as the limit of f(x) at x = a.
Limits Detailed Video Explanation
Properties of Limits
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The following are the properties and theorems related to the concept of limits.
- Sum rule: The limit of the sum of two functions is equal to the sum of the limits of their individual functions. The sum rule can be represented by: Limx→a [f(x) + g(x)] = limx->a f(x) + limx->a g(x).
- Difference rule: Limit of difference of two functions is equal to the difference of limits of their individual functions. The difference rule can be represented by: Limx→a [f(x) - g(x)] = limx->a f(x) - limx->a g(x).
- Product rule: The Limit of product of two functions is equal to the product of the limits of their individual functions. The product rule can be represented by: Limx→a [f(x)*g(x)] = limx->a f(x)*limx->a g(x).
- Quotient rule: The limit of the quotient of any two functions is equal to the limit of the quotient of both functions when the denominator is not equal to zero. The quotient rule can be represented by: Limx→a f(x)/g(x) = limx->a f(x)/limx->a g(x)
- Power rule: The power of any root function is stated as: Limx→a√|f(x)| = √limx->a |f(x)|. This is an effective approach in solving power roots.
- Positive Integers: For any positive integers n, Lim xn – an/x-a = na(n-1).
- Sandwich Theorem: Let us assume that f, g, h are real functions such that f(x) ≤ g(x) ≤ h(x) then for any real number a, if limx→a f(x) = l = limx->a g(x)
- then, limx→a g(x) = l.
- L-Hospital Rule: The L-Hospital rule enables you to solve a function by splitting it into individual limits. It is represented as: limx→a f(x)/g(x) = limx->a f1(x)/g1(x)
What is Limit Formula?
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The limit formula is typically used in order to calculate the derivative of a function.
- The limit can be defined as the value of the function which is seen to approach just as the input approaches the given value.
- Limits are typically used for making approximations of the calculation that are close to the actual value of the quantity.
Calculation of LimitCalculation of a value if Limits of a Function are given in a Quotient Form: If the limit of a function has been given in the form of a quotient, the expression can be calculated by the factoring method. The steps to follow for the same include:
How can I evaluate the Limits of a Quotient? The limit of a quotient can be calculated by predicting the LCD of the expression. The steps for the same are:
How can I Evaluate the Limit of a Function that Contains a Root? For this case,
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Limit formulas
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Here are some basic limit formulas 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:
- \(\begin{array}{l} \lim_{x\to 0}(1+x)^{\frac{1}{x}}=e\end{array}\)
- \(\begin{array}{l} \lim_{x\to \infty }(1+\frac{1}{x})^{x}=e\end{array}\)
- \(\begin{array}{l}\lim_{x\to \infty }(1+\frac{a}{x})^{x}=e^{a}\end{array}\)
Also Read:
Logarithm and Exponential Formulas
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Some of the logarithm and exponential formulas are listed below.
- \(\begin{array}{l}\lim_{x\to 0 }e^{x}=1\end{array}\)
- \(\lim_{x\to 0 }\frac{e^{x}-1}{x}=1\)
- \(\begin{array}{l}\lim_{x\to 0 }\frac{a^{x}-1}{x}=log_{e}\: a\end{array}\)
- \(\begin{array}{l}\lim_{x\to 0 }\frac{\log (1+x)}{x}=1\end{array}\)
- \(\begin{array}{l}\lim_{x\to a }\frac{x^{n}-a^{n}}{x-a}=na^{n-1}\end{array}\)
Discover about the Chapter video:
Continuity and Differentiability Detailed Video Explanation:
Things to Remember
- In mathematics, limit simply gives the near values or high values of the output.
- Limits are essential to determine derivatives, continuities, and integrals of functions.
- Sum rule, product rule, difference rule, and quotient rule constitute the properties of limits.
- Limits can be of different types such as left-hand side limits, right-hand side limits, infinite limits and one-sided infinite limits.
- The L-Hospital rule enables you to solve a function by splitting it into individual limits.
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 …
Sample Questions
Ques. Solve limx->0 (sin x)/x using L-Hospital Rule. (2 marks)
Ans. limx->0 (sin x)/x = 0/0
Using L-Hospital’s rule,
limx->0 (sin x)/x = limx->0 (cos x)/1
limx->0 (cos x) = 1 and limx->0 (1) = 1
limx->0 (sin x)/x = 1
Ques. What is the use of the L-Hospital Rule? (1 mark)
Ans. L-Hospital Rule is used to solve problems whose numerator and denominator become zero by substituting limits mostly for indeterminant problems.
Ques. State and prove L-Hospitals rule? (2 marks)
Ans. limx->0 f(x)/g(x) = limx->c f(x)-0/g(x)-0
→ limx->c f(x)-f(c)/g(x)-g(c)
→ limx->c ((f(x)-f(c)/x-c)/(g(x)-g(c)/x-c
→ limx->c ((f(x)-f(c)/x-c)/ limx->c ((f/g(x)-f/g(c)/x-c)
→ limx->c f1(c)/g1(c)
Hence proved.
Ques. Who developed the idea of limits? (1 mark)
Ans. Archimedes developed the idea of limits which is to measure curved surfaces which later became an integral part of the calculus.
Ques. Can multiple limits exist for the same function if not explain why? (1 mark)
Ans. A Limit cannot have multiple functions because no function can exist between two different time intervals at the same time.
Ques. What are the cases in which limit doesn’t exist? (2 marks)
Ans. 1)If the graph has a gap at x value a then limit doesn’t exist
2)If the function doesn’t have a finite value
3) if the function value tends to zero.
Ques. Evaluate lim x->3 (2x3 – 3x2 +1) (2 marks)
Ans. This can written in the form of ,
=lim x->3 (2x3)- lim x->3 (3x2)+ lim x->3 (1)
= 2lim x->3 (x3)- 3lim x->3 (x2)+ (1)
= 2(33)- 3(32)+1
= 2*27- 3*9+1
= 28
Hence the value of the above given quadratic equation is 3.
Ques. Evaluate the given limit lim x->0(1+cosx)2 secx (3 marks)
Ans. Let us assume y= cosx,
Since x->0 then y->0
=Sec x = 1/cos x = 1/y
= lim x->0(1+cosx)2 secx ---------→ (1)
= lim y->0(1+y)2/y
According to the formula lim x->0 (1+x)1/x = e
Substitute this formula in equation (1) then lim y->0(1+y)2/y
=e2
Ques. Evaluate the following limit lim x-> 0 (ex - e-x) / sin x (4 marks)
Ans. lim x-> 0 (ex - e-x) / sin x
= lim x-> 0 (ex - (1/ex)) / sin x
= lim x-> 0 ((ex)2 - 1)/ex sin x
= lim x-> 0 (e2x - 1)/ex sin x
Now we are going to multiply numerator by 2x/2x and denominator sin x by (x/x)
= lim x-> 0 (e2x - 1)(2x/2x)/ex sin x (x/x)
= lim x-> 0 ((e2x - 1)/2x)(2x/x)/(ex (sin x/x))
= 2lim x-> 0 ( (e2x - 1)/2x) / lim x-> 0 ( ex lim x-> 0(sin x/x))
= 2(1)/1(1)
= 2
Hence, the value of lim x-> 0 (ex - e-x) / sin x is 2.
Ques. Evaluate limx->1log(x+1) (2 marks)
Ans. let us assume as x=1 in log(x+1)
= limx->1log(x+1)
=log(1+1)
=log2
Ques. Evaluate limx->0(1+x)1/3x (2 marks)
Ans. This can be written in the form of limx->0(1+x)1/3x
= limx->0(1+x)1/3x
= limx->0[(1+x)1/x]1/3
=e1/3
Hence the evaluation of limx->0(1+x)1/3x is e1/3
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