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Limit of function in function of limit of sequence of values of function

Source: RomNO 2003, grade xi. p. 3

August 27, 2019
functioncontinuitylimitsreal analysisromania

Problem Statement

Let be two functions f,g:R≥0⟶R f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} having that properties that f f is continuous, g g is nondecreasing and unbounded, and for any sequence of rational numbers (xn)n≥1 \left( x_n \right)_{n\ge 1} that diverges to ∞, \infty , we have 1=lim⁡n→∞f(xn)g(xn). 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . Prove that 1=lim⁡x→∞f(x)g(x).1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) .
Radu Gologan
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