If $f(x) = \begin{cases} mx^2 + n, & x < 0 \\ nx + m, & 0 \leq x \leq 1 \\ nx^3 + m, & x > 1 \end{cases}$
For what integers $m$ and $n$ do $\lim_{x \to 0} f(x)$ and $\lim_{x \to 1} f(x)$ exist?
For what integers $m$ and $n$ do $\lim_{x \to 0} f(x)$ and $\lim_{x \to 1} f(x)$ exist?
(A) For $\lim_{x \to 0} f(x)$ to exist,the left-hand limit and right-hand limit at $x = 0$ must be equal.
$\lim_{x \to 0^-} f(x) = \lim_{x \to 0} (mx^2 + n) = m(0)^2 + n = n$
$\lim_{x \to 0^+} f(x) = \lim_{x \to 0} (nx + m) = n(0) + m = m$
Thus,$\lim_{x \to 0} f(x)$ exists if $m = n$.
For $\lim_{x \to 1} f(x)$ to exist,the left-hand limit and right-hand limit at $x = 1$ must be equal.
$\lim_{x \to 1^-} f(x) = \lim_{x \to 1} (nx + m) = n(1) + m = n + m$
$\lim_{x \to 1^+} f(x) = \lim_{x \to 1} (nx^3 + m) = n(1)^3 + m = n + m$
Since $n + m = n + m$,the limit $\lim_{x \to 1} f(x)$ exists for all integers $m$ and $n$.
Therefore,the condition for both limits to exist is $m = n$ for any integer values of $m$ and $n$.
$\lim_{x \to 0^-} f(x) = \lim_{x \to 0} (mx^2 + n) = m(0)^2 + n = n$
$\lim_{x \to 0^+} f(x) = \lim_{x \to 0} (nx + m) = n(0) + m = m$
Thus,$\lim_{x \to 0} f(x)$ exists if $m = n$.
For $\lim_{x \to 1} f(x)$ to exist,the left-hand limit and right-hand limit at $x = 1$ must be equal.
$\lim_{x \to 1^-} f(x) = \lim_{x \to 1} (nx + m) = n(1) + m = n + m$
$\lim_{x \to 1^+} f(x) = \lim_{x \to 1} (nx^3 + m) = n(1)^3 + m = n + m$
Since $n + m = n + m$,the limit $\lim_{x \to 1} f(x)$ exists for all integers $m$ and $n$.
Therefore,the condition for both limits to exist is $m = n$ for any integer values of $m$ and $n$.
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