Chapter 1: Q. 85 (page 151)

Use limits to prove that the limits of a polynomial $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{1} x + a_{0}$ are the same as the limits of its leading term $a_{n} x^{n}$ as $x \to \infty$ and as $x \to - \infty$ . (Hint: Show that $\lim_{x \to \infty} f (x)$ is equal to $\lim_{x \to \infty} a_{n} x^{n}$ by factoring out $a_{n} x^{n}$ from $f (x)$ .)

Short Answer

Expert verified

The limits of a polynomial $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{1} x + a_{0}$ are the same as the limits of its leading term $a_{n} x^{n}$ as $x \to \infty$ and as $x \to - \infty$ .

Step by step solution

Step 1. Given Information

We are given a function,

$f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{1} x + a_{0}$

Step 2. Proving the statement

Take limit in the function as below,

$\lim_{x \to \infty} (a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}) = \lim_{x \to \infty} (a_{n} x^{n} (1 + \frac{a_{n - 1}}{a_{n} x} + \dots + \frac{a_{1}}{a_{n} x^{n - 1}} + \frac{a_{0}}{a_{0} x^{n}})) = (\lim_{x \to \infty} a_{n} x^{n}) (1 + 0 + \dots + 0 + 0) = \lim_{x \to \infty} a_{n} x^{n}$

Similarly for $\lim_{x \to - \infty} f (x)$ .

Hence, limits of a polynomial $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{1} x + a_{0}$ are the same as the limits of its leading term $a_{n} x^{n}$ as $x \to \infty$ and as $x \to - \infty$ .

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Use limits to prove that the limits of a polynomial f(x)=anxn+an-1xn-1+a1x+a0 are the same as the limits of its leading term anxn as x→∞ and as x→-∞. (Hint: Show that limx→∞f(x) is equal to limx→∞anxn by factoring out anxn from f(x).)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

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