I need help with this problem and an explanation of this problem. ### Understanding the Limit of a Polynomial FunctionTo show that for any polynomial \( p \), the limit of \( p(x) \) as \( x \) approaches \( a \) equals \( p(a) \).**Theorem Statement:**If \( p \) is a polynomial, show that \[ \lim_{{x \to a}} p(x) = p(a). \]**Proof:**1. **Expression of a Polynomial:** Since \( p(x) \) is a polynomial, it can be written as: \[ p(x) = a_0 + a_1(x - a) + a_2(x - a)^2 + \cdots + a_n(x - a)^n. \] 2. **Limit of the Polynomial:** We take the limit of both sides as \( x \) approaches \( a \): \[ \lim_{{x \to a}} p(x) = \lim_{{x \to a}} \left( a_0 + a_1(x - a) + a_2(x - a)^2 + \cdots + a_n(x - a)^n \right). \] 3. **Limit Laws Application:** By applying the limit laws, we can split the limit of a sum into the sum of the limits: \[ = \lim_{{x \to a}} \left( a_0 \right) + \lim_{{x \to a}} \left( a_1(x - a) \right) + \lim_{{x \to a}} \left( a_2(x - a)^2 \right) + \cdots + \lim_{{x \to a}} \left( a_n(x - a)^n \right). \] 4. **Evaluating Each Term:** Each term can be evaluated separately: \[ = a_0 + a_1 \lim_{{x \to a}} (x - a) + a_2 \lim_{{x \to a}} (x - a)^2 + \cdots + a_n \lim_{{x \to a}} (x - a)^n. \] 5. **Substituting the Limits:** We know that for any integer \( k \geq 1 \), \(\lim_{{x \to a

If is a polynomial, we have to show that .

If p is a polynomial, show that lim p(x) = p(a). x→a Since p(x) is a polynomial, p(x) = a + a₁ lim p(x) x→a ²√(²0+² = lim x→a a₁ lim = 30 +₁ = a + a₁ x→a ( [ Thus, for any polynomial p, lim_ p(x) = p(a). + a₂x² +. +²₁x²) + a₂ lim (x²) + ... + a lim (x²) x→a x→a +anan = p(a). + a₂a²+ +... + ax^. Thus, by the limit laws, ++

If p is a polynomial, show that lim p(x) = p(a). x→a Since p(x) is a polynomial, p(x) = a + a₁ lim p(x) x→a ²√(²0+² = lim x→a a₁ lim = 30 +₁ = a + a₁ x→a ( [ Thus, for any polynomial p, lim_ p(x) = p(a). + a₂x² +. +²₁x²) + a₂ lim (x²) + ... + a lim (x²) x→a x→a +anan = p(a). + a₂a²+ +... + ax^. Thus, by the limit laws, ++

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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