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Algebra of Limits - Practice Questions & MCQ

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

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  • Algebra of Limits is considered one of the most asked concept.

  • 54 Questions around this concept.

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QuestionMEDIUM

The value of the \lim _{t \rightarrow 1} \frac{t-t^2}{\sqrt{1+\log _2 t}-1} is equal to:

 

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Using the power law for limits evaluate \lim_{x\rightarrow -2}\left ( \frac{x^{3}-2}{x^{3}+14x^{2}+2} \right )^{2}.

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The set of all values of a for which \lim _{x \rightarrow a}([x-5]-[2 x+2])=0 where [\propto] denotes the greatest integer less than or equal to \alpha is equal to

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Let f, g and h be the real valued functions defined on \mathbb{R} as
f(x)=\left\{\begin{array}{cl}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{array}, g(x)=\left\{\begin{array}{cc}\frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\ 1, & x=-1\end{array}\right.\right.
and \mathrm{h}(\mathrm{x})=2[\mathrm{x}]-\mathrm{f}(\mathrm{x}),where [\mathrm{x}] is the greatest integer \leq \mathrm{x}.
Then the value of \lim _{x \rightarrow 1} g(h(x-1)) is

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If \alpha>\beta>0 \ \text{are the roots of the equation a }x^2+b x+1=0, and \lim _{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^2+b x+a\right)}{2(1-a x)^2}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right), \text{then k is equal to}

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Let \mathrm{a_{n+1}=\sqrt{\left(2+a_n\right)}, n=1,2,3, \ldots \ldots\: and\: \: a_1=3}, then the value of \mathrm{\lim _{n \rightarrow a} a_n}  is............

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If \mathrm{f(a)=2, f^{\prime}(a)=1, g(a)=-1, g^{\prime}(a)=2}, then the value of \mathrm{\lim _{x \rightarrow a} \frac{g(x) f(a)-g(a) f(x)}{x-a}} is equal to .......

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If \mathrm{f(x)=\cot ^{-1}\left(\frac{2 x}{1-x^2}\right) and \: g(x)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), then\: \: \lim _{x \rightarrow a} \frac{f(x)-f(a)}{g(x)-g(a)}\left(0<a<\frac{1}{2}\right)} is ......

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Let f: R \rightarrow[0, \infty) be such that \lim _{x \rightarrow 5} f(x) exists and \lim _{x \rightarrow 5} \frac{f(x)^2-9}{\sqrt{|x-5|}}=0 Then \lim _{x \rightarrow 5} f(x) equals to :

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Concepts Covered - 1

Algebra of Limits

Algebra of Limits

Let f(x) and g(x) be defined for all x ≠ a over some open interval containing a. Assume that L and M are real numbers such that \lim _{x \rightarrow a} f(x)=L   and \lim _{x \rightarrow a} g(x)=M. Let c be a constant. Then, each of the following statements hold:
 

\\\mathrm{ { \mathbf{Sum \;law \;for \;limits}: }} \lim _{x \rightarrow a}(f(x)+g(x))=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x)=L+M\\\\\mathrm { \mathbf{Difference\;law\; for\; limits: }} \lim _{x \rightarrow a}(f(x)-g(x))=\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)=L-M\\\\\mathrm { \mathbf{Constant\; multiple\; law \;for\; limits:} } \lim _{x \rightarrow a} c f(x)=c \cdot \lim _{x \rightarrow a} f(x)=c L\\\\\mathrm { \mathbf{Product \;law\; for \;limits:} } \lim _{x \rightarrow a}(f(x) \cdot g(x))=\lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)=L \cdot M


\\\mathrm { \mathbf{Quotient\; law \;for \;limits: }} \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}=\frac{L}{M} \text { for } M \neq 0\\\\\\\mathrm { \mathbf{Power \;law \;for \;limits:} } \lim _{x \rightarrow a}(f(x))^{n}=\left(\lim _{x \rightarrow a} f(x)\right)^{n}=L^{n} \text { for every positive integer } n\\\\\mathrm{\mathbf{Composition\;law\;of\;limit:}}\lim_{x\rightarrow a}(fog)(x)=f\left ( \lim_{x\rightarrow a}g(x) \right )=f(M), \text{only if f(x) is }\\\text{continuous at g(x)=M.}\\\\\mathrm{\mathbf{If}\;{lim_{x\rightarrow a}f(x)=+\infty}\;\mathbf{or}-\infty,\;\mathbf{then}\;\lim_{x\rightarrow a}\frac{1}{f(x)}=0}

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Algebra of Limits

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Algebra of Limits Current Topic

Limit
Left-Hand Limits and Right-Hand Limits
Algebra of Limits
Limit of Indeterminate Form and Algebraic limit
Limit of Algebraic function
Limit Using Expansion
Sandwich Theorem
Limits of Trigonometric Functions
Exponential and Logarithmic Limits
Limits of the form (1 power infinity)

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Algebra of Limits

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