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    Sandwich Theorem - Practice Questions & MCQ

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

    Quick Facts

    • 65 Questions around this concept.

    Solve by difficulty

    QuestionMEDIUM

    For each $t \epsilon R$, let $[t]$ be the greatest integer less than or equal to $t$. Then

    $
    \lim _{x \rightarrow 0} x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\ldots .+\left[\frac{15}{x}\right]\right)
    $
     

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    \text { If } \lim _{x \rightarrow 0} \frac{a e^x+b \cos x+c e}{e^{2 x}-2 e^x+1}=4 \text {, then }:

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    \text { Evaluate: } \lim _{x \rightarrow 0}\left(\frac{1^x+2^x+3^x+\ldots+n^x}{n}\right)^{a / x}:

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    Which of the following is true for the real valued functions f(x),h(x),g(x) 

    \lim _{x \rightarrow a} f(x)=L, \lim _{x \rightarrow-a} g(x)=L, \lim _{x \rightarrow-a} f(x)=N, \lim _{x \rightarrow a} g(x)=N?

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    The integer \mathrm{n} for which \mathrm{\lim _{x \rightarrow 0}\left\{\frac{(\cos x-1)\left(\cos x-e^2\right)}{x^n}\right\}} is a finite, non-zero real number, is

     

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    \mathrm{\lim _{n \rightarrow \infty}\left[\sqrt{\left(n^2+n+1\right)}-\left[\sqrt{\left(n^2+n+1\right)}\right]\right)}
    where [ ] denotes the greatest integer function:

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    \mathrm{\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{r}{\left(n^2+n+r\right)} \text { equals : }}

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    \mathrm{ Assume\, \, that \lim _{\theta \rightarrow-1} f(\theta) exists and \frac{\theta^2+\theta-2}{\theta+3} \leq \frac{f(\theta)}{\theta^2} \leq \frac{\theta^2+2 \theta-1}{\theta+3} }
    \mathrm{ holds\, \, for \, \, certain\, \, interval\, \, containing\, \, the \, \, point \, \, \theta=-1 then \lim _{\theta \rightarrow-1} f(\theta) : }

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    \text{ Evaluate :}\mathrm{\lim _{n \rightarrow \infty} \frac{1}{1+n^2}+\frac{2}{2+n^2}+\ldots .+\frac{n}{n+n^2}}.

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    \text{ Evaluate :}\mathrm{\lim _{x \rightarrow \infty} \frac{x+7 \sin x}{-2 x+13}}  using Sandwich Theorem.

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    Concepts Covered - 1

    Sandwich Theorem

    Sandwich Theorem

    Sandwich theorem is also known as the squeeze play theorem. It is typically used to find the limit of a function via comparison with two other functions whose limits are known or are easily calculated.

    Sandwich Theorem 

    Let $f(x), g(x)$ and $h(x)$ be real functions such that $f(x) \leq g(x) \leq h(x)$ for all $x$ in the neighbourhood of $x=a$. If $\lim _{x \rightarrow a} f(x)=l=\lim _{x \rightarrow a} h(x)$, then $\lim _{x \rightarrow a} g(x)=l$.

     

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    Sandwich Theorem

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    Sandwich Theorem 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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    Reference Books

    Sandwich Theorem

    Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

    Page No. : 2.7

    Line : 6

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