Question

Compute the above limit function ​

Answer 1

Answer:

• 4

• 3

Step-by-step explanation:

$\lim_{x \rightarrow 2^+}([x] + x) \\ => 2 +2 // => 4 \\ \lim_{x \rightarrow 2^-}([x]+x) \\ => 1 + 2 \\ = 3 `$

[.] denotes the Gretatest Integer Function(GIF) This means if X(input) lies in [n, n+1), then the Greatest Integer Function of X(output) will be n.

When x-> 2⁻,

the value will be somewhat like 1.99999 that is slightly smaller than 2, so when we're talking about slightly smaller than 2, the value comes under the range [1, 2)

therefore, the value of [x] will be 1.

When x-> 2⁺, it's value will be somewhat like 2.0000001, so that the value lies In the range [2, 3)

therefore, the value of [x] will be 2.

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Hope this helps!

Answer 2

We know that, if a is an integer,

$\small\longrightarrow x\in[a,\ a+1)\quad\iff\quad[x]=a$

Then we see that,

$\small\longrightarrow x\in[1,\ 2)\quad\iff\quad[x]=1$

$\small\longrightarrow x\in[2,\ 3)\quad\iff\quad[x]=2$

Thus the function $\smallf:[1,\ 3)\to\mathbb{R}$ defined by $\smallf(x)=[x]+x$ is such that,

$\small\text{ \longrightarrow f(x)=\left\{\begin{array}{ll}1+x,&1\leq x<2\\2+x,&2\leq x<3\end{array}\right. }$

Hence,

$\small\displaystyle\longrightarrow\lim_{x\to2^+}f(x)=\lim_{x\to2^+}(2+x)$

$\small\displaystyle\longrightarrow\lim_{x\to2^+}f(x)=2+2$

$\small\text{ \displaystyle\longrightarrow\underline{\underline{\lim_{x\to2^+}f(x)=4}} }$

and,

$\small\displaystyle\longrightarrow\lim_{x\to2^-}f(x)=\lim_{x\to2^-}(1+x)$

$\small\displaystyle\longrightarrow\lim_{x\to2^-}f(x)=1+2$

$\small\text{ \displaystyle\longrightarrow\underline{\underline{\lim_{x\to2^-}f(x)=3}} }$