Question

If f(x) = \(f(x)=\left\{\begin{matrix} mx^2+n, &\,\,\,\,x<0 \\   nx+m,&\,\,\,\, 0\leq x\leq1 \\   nx^3+m,&\,\,\,\, x>1  \end{matrix}\right.\). For what integers m and n does both \(\lim_{x\rightarrow 1}\)f(x) and \(\lim_{x\rightarrow 1}\) f(x) exist?

Answer 1

Given:

The function is defined as:

f(x) =

⎧ mx² + n,    x < 0
⎨ nx + m,    0 ≤ x ≤ 1
⎩ nx³ + m,    x > 1

We are required to find integers m and n such that both

lim_x→0 f(x) and lim_x→1 f(x) exist.

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Step 1: Condition for existence of lim_x→0 f(x)

Left hand limit at x = 0:

lim_x→0⁻ f(x) = m(0)² + n = n

Right hand limit at x = 0:

lim_x→0⁺ f(x) = n(0) + m = m

For the limit to exist,

n = m   ...(1)

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Step 2: Condition for existence of lim_x→1 f(x)

Left hand limit at x = 1:

lim_x→1⁻ f(x) = n(1) + m = n + m

Right hand limit at x = 1:

lim_x→1⁺ f(x) = n(1)³ + m = n + m

Since both limits are equal automatically, lim_x→1 f(x) exists for all m and n.

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Step 3: Combine conditions

From Step 1:

m = n

This ensures the existence of both limits.

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Final Answer:

Both lim_x→0 f(x) and lim_x→1 f(x) exist
when m = n, where m and n are integers.