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Left-Hand Limits and Right-Hand Limits - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • 69 Questions around this concept.

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Under which of the following conditions does the limit of the function shown as  \lim _{x \rightarrow p} f(x)  exist?

\lim _{x \rightarrow \pi} \frac{\cos ^{-1}(\cos x)-\pi}{\sin ^{-1}(\sin x)}  is equal to.


 

If \mathrm{[x]} denotes the largest integer that is less than or equal to \mathrm{x}, then \mathrm{\lim _{x - 3}[x]} is
 

If \mathrm{[x]} denotes the greatest integer less than or equal to \mathrm{x, and \: \: f(x)=[x]+[-x]}, then \mathrm{\lim _{x \rightarrow 0} f(x)} is
 

If \mathrm{f(x)=[x]+[-x],} then  \mathrm{\lim_{x\rightarrow 2}f(x)} is

If  \mathrm{f(x)=[x]+[-x]}, then \mathrm{\lim _{x \rightarrow 1 / 2} f(x)} is
 

\lim_{x\rightarrow 3}\mathrm{(2-[-x])} is

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\mathrm{\lim _{x \rightarrow 1}\left(1-\left[\frac{-x}{2}\right]\right)} is
 

For the function\mathrm{ f(x)=\lim _{n \rightarrow x}\left\{\frac{\ln (x+2)-x^{2 n} \cdot \sin x}{1+x^{2 n}}\right\}, }which of the following is true?
 

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\mathrm{{x}} denotes the fractional part of a real no. \mathrm{x}, then \mathrm{\lim _{x \rightarrow 0}\left\{\frac{\ln (1+\{x\})}{\{x\}}\right\}} is
 

Concepts Covered - 1

Left-Hand Limits and Right-Hand Limits

Left-Hand Limits and Right-Hand Limits

With continuing from previous concept, we can approach the input of a function from either side of a value—from the left or the right.  

we had our function as

 f(x)=\frac{(x+1)(x-7))}{(x-7)}, x \neq 7 which becomes equivalent to the function

f(x)=x+1, x \neq 7

now, let us observe the values of f(x) nearby x = 7.


 

Approaching 7 “from the left” means that the values of input are just less than 7. And if for such values of x, the values of f(x) are close to L, then L is called the left-hand limit of function at x = 7. For this function, 8 is the left-hand limit of the function f(x) at x = 7.

Approaching 7 “from the right” means that the values of input are just larger than 7. And if for such values of x, the values of f(x) are close to R, then R is called the right-hand limit of function at x = 7. For this function, 8 is the right-hand limit of the function f(x) at x = 7.

To indicate the left-hand limit, we write \lim _{x \rightarrow 7^{-}} f(x)=8. 7- indicates the values that less than 7 and are infinitesimally close to 7

To indicate the right-hand limit, we write\lim _{x \rightarrow 7^{+}} f(x)=8. 7+ indicates the values that greater than 7 and are infinitesimally close to 7

The left-hand and right-hand limits are the same for this function.

( It denotes that if we can redefine the function, it is possible to define value of function at x = 7 as f(7) = 8... this is the scenario we wiil be observing the the chapter again and again where, we define it that if LHL = RHL at a point x=a, then Limit of function at the point can be defined)

The left-hand limit of a function f(x) as x approaches a from the left is denoted by \lim _{x \rightarrow a^{-}} f(x)=LHL

The right-hand limit of a function f(x) as x approaches a from the right is denoted by \lim _{x \rightarrow a^{+}} f(x)=RHL

 

Now consider a function, \\\mathrm{f(x)=\frac{|x|}{x}} 

Let’s check the behavior of f(x) in the neighborhood of x = 0

\\\mathrm{LHL=\lim_{x\rightarrow 0^{-}}\frac{|x|}{x}} \\\\\text{As x is just less than 0, we can replace it by (0 - h), where h is positive and very close to 0} \\\\\mathrm{\quad\quad\;=\lim_{h\rightarrow 0^{+}}\frac{|0-h|}{0-h}}\\\\\mathrm{So,\;we\;have\;}\\\mathrm{\quad\quad\;=\lim_{h\rightarrow 0^{+}}\frac{|-h|}{-h}}\\\\\mathrm{\quad\quad\;=\lim_{h\rightarrow 0^{+}}\frac{h}{-h}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;[\because |-h|=h]}\\\\\mathrm{\quad\quad\;=-1}\\\\\mathrm{RHL=\lim_{x\rightarrow 0^{+}}\frac{|x|}{x}}\\\\\mathrm{\quad\quad\;=\lim_{h\rightarrow 0^{+}}\frac{|0+h|}{0+h}}\\\\\mathrm{\quad\quad\;=\lim_{h\rightarrow 0^{+}}\frac{h}{h}=1}\\\text{Here, we have \;RHL}\neq\text{LHL} 

 

Existence of a limit of a function

From the above example, we can define the existence of limit

\\\mathrm{The\;limit\;of \;a\;function\;\mathit{f(x)}\;at\;\mathit{x=a}\;exists\;if\;\mathit{\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)}}\\\mathrm{or\;\mathit{\lim_{h\rightarrow 0^+}f(a-h)=\lim_{h\rightarrow 0^+}f(a+h)}}.

i.e., LHL = RHL at x = a 

Also notice that the limit of a function can exist even when f(x) is not defined at x = a.

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Left-Hand Limits and Right-Hand Limits

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 2.1

Line : 20

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