Basic Analysis I — Real Analysis · Problem 3.5.6

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · limits-at-infinity

Problem

Suppose P(x):=xn+an−1xn−1+⋯+a1x+a0P(x) := x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 is a monic polynomial of degree n≥1n \geq 1 (monic means that the coefficient of xnx^n is 1). a) Show that if nn is even, then lim⁡x→∞P(x)=lim⁡x→−∞P(x)=∞\lim_{x \to \infty} P(x) = \lim_{x \to -\infty} P(x) = \infty. b) Show that if nn is odd, then lim⁡x→∞P(x)=∞\lim_{x \to \infty} P(x) = \infty and lim⁡x→−∞P(x)=−∞\lim_{x \to -\infty} P(x) = -\infty (see previous exercise).

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