If f(x) = * 20 c m x^2 + n, nx + m, n x^3 + m, . * 20 c x 1 . For what integers m and n does both _ x 0 f(x) and _ x 1 f(x) exist?

It is given that _ x 0 f(x) and _ x 1 f(x) both exist. ; _ x 0^ - f(x) and = _ x 0^ + f(x) = _ x 1^ + f(x) Now _ x 0^- f(x) = _ x 0 (mx^2+ n) = n _ x 0^ + f(x) = _ x 0 (nx + m) = m Now _ x 0^ + f(x) = _ x 0^ + f(x) n = m ...(i) For _ x 0 f(x) to exist we need m = n Also, _ x 1^ - f(x) = _ x 1 (nx + m) = n + m _ x 1^ + f(x) = _ x 1 (nx^3 + m) = n + m Now _ x 1^ - f(x) = _ x 1^ + f(x) n + m = n + m Thus _ x 0 f(x) exists for any integral value of m and n.

If f(x) = * 20 c m x^2 + n, nx + m, n x^3 + m, . * 20 c x < 0 0 x…

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Question

If $f(x) = \left\{ {\begin{array}{*{20}{c}} {m{x^2} + n,} \\ {nx + m,} \\ {n{x^3} + m,} \end{array}} \right.\begin{array}{*{20}{c}} {x < 0} \\ {0 \le x \le 1} \\ {x > 1} \end{array}$ .
For what integers m and n does both $\mathop {\lim }\limits_{x \to 0} f(x)$ and $\mathop {\lim }\limits_{x \to 1} f(x)$ exist?

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Answer

It is given that
$\mathop {\lim }\limits_{x \to 0}$ f(x) and $\mathop {\lim }\limits_{x \to 1}$ f(x) both exist.
$\Rightarrow \;\mathop {\lim }\limits_{x \to {0^ - }}$ f(x) and = $\mathop {\lim }\limits_{x \to {0^ +}}$f(x) = $\mathop {\lim }\limits_{x \to {1^ + }}$ f(x)
Now $\mathop {\lim }\limits_{x \to {0^- }}$ f(x) = $\mathop {\lim }\limits_{x \to {0 }} (mx^2+ n) = n$
$\mathop {\lim }\limits_{x \to {0^ + }}$ f(x) = $\mathop {\lim }\limits_{x \to {0 }}$ (nx + m) = m
Now $\mathop {\lim }\limits_{x \to {0^ + }}$ f(x) = $\mathop {\lim }\limits_{x \to {0^ + }}$ f(x) $\Rightarrow$ n = m ...(i)
For $\mathop {\lim }\limits_{x \to 0}$ f(x) to exist we need m = n
Also, $\mathop {\lim }\limits_{x \to {1^ - }}$ f(x) = $\mathop {\lim }\limits_{x \to {1}} (nx + m) = n + m$
$\mathop {\lim }\limits_{x \to {1^ + }}$ f(x) = $\mathop {\lim }\limits_{x \to {1}} (nx^3 + m) = n + m$
Now $\mathop {\lim }\limits_{x \to {1^ - }}$ f(x) = $\mathop {\lim }\limits_{x \to {1^ + }}$ f(x) $\Rightarrow$ n + m = n + m
Thus $\mathop {\lim }\limits_{x \to 0}$ f(x) exists for any integral value of m and n.

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