(1) Evaluate the following limits. Justify your answers. (a) lim, (b) lim, (c) lim,x(x-v=) (d) lim (V + 12z – z) (e) lim 12- (2) Use the limit definition of the derivative to find I+ Do not use the power rule or the chain rule. Your answer should start with +1= lim h-0 (3) Find an equation of the tangent line to the curve 1 at the point (2, }).
(1) Evaluate the following limits. Justify your answers. (a) lim, (b) lim, (c) lim,x(x-v=) (d) lim (V + 12z – z) (e) lim 12- (2) Use the limit definition of the derivative to find I+ Do not use the power rule or the chain rule. Your answer should start with +1= lim h-0 (3) Find an equation of the tangent line to the curve 1 at the point (2, }).
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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Question
![(1) Evaluate the following limits. Justify your answers.
(a) \(\lim_{x \to -\infty} \frac{8x^2 + 2}{x + 4}\)
(b) \(\lim_{x \to \infty} \frac{5x^2 - x}{x + 4}\)
(c) \(\lim_{x \to \infty} (x - \sqrt{x^2})\)
(d) \(\lim_{x \to \infty} (\sqrt{x^2 + 12x} - x)\)
(e) \(\lim_{x \to \infty} \frac{12x^2 - 5x \cdot 3^x}{6 \cdot 4^x + 6 \cdot 3^x}\)
(2) Use the limit definition of the derivative to find
\[
\frac{d}{dx} \sqrt{x^2 + 1}
\]
Do not use the power rule or the chain rule. Your answer should start with
\[
\frac{d}{dx} \sqrt{x^2 + 1} = \lim_{h \to 0} \ldots
\]
(3) Find an equation of the tangent line to the curve
\[
y = \frac{1}{x^2 + 1}
\]
at the point \((2, \frac{1}{5})\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1300845a-828a-484d-8f06-319293900125%2F7ac0c855-2f81-4f03-9a00-130c971e2842%2Fc674yj3_processed.png&w=3840&q=75)
Transcribed Image Text:(1) Evaluate the following limits. Justify your answers.
(a) \(\lim_{x \to -\infty} \frac{8x^2 + 2}{x + 4}\)
(b) \(\lim_{x \to \infty} \frac{5x^2 - x}{x + 4}\)
(c) \(\lim_{x \to \infty} (x - \sqrt{x^2})\)
(d) \(\lim_{x \to \infty} (\sqrt{x^2 + 12x} - x)\)
(e) \(\lim_{x \to \infty} \frac{12x^2 - 5x \cdot 3^x}{6 \cdot 4^x + 6 \cdot 3^x}\)
(2) Use the limit definition of the derivative to find
\[
\frac{d}{dx} \sqrt{x^2 + 1}
\]
Do not use the power rule or the chain rule. Your answer should start with
\[
\frac{d}{dx} \sqrt{x^2 + 1} = \lim_{h \to 0} \ldots
\]
(3) Find an equation of the tangent line to the curve
\[
y = \frac{1}{x^2 + 1}
\]
at the point \((2, \frac{1}{5})\).
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