QUESTION#2PART (A)Evaluate the following limits:i) lim Jx +4-21x' +x? - 5x +3ii) limx' - 3x+2x --04PART (B)1-cos9hEvaluate the following limits: i.limocosóh-1ii lim,a(Vx² + 4x – vx²+ 3x)

Answered: PART (A) Evaluate the following limits:…

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PART (A) Evaluate the following limits: i) lim J +4-21 x' +x - 5x +3 x'-3x+2 ii) lim 4

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 42E

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

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(i)

$\underset{x \to 0}{l i m} \frac{\sqrt{x^{2} + 4} - 2}{x^{2}}$

The form is of $\frac{0}{0}$

Let's use the L'Hsopital rule,

$\begin{array}{rcl} \underset{x \to 0}{l i m} \frac{\frac{1}{2}}{\sqrt{x^{2} + 4}} \times \frac{2 x}{2 x} & = & \underset{x \to 0}{l i m} \frac{1}{2} \frac{1}{\sqrt{x^{2} + 4}} \\ = & \frac{1}{2} \times \frac{1}{2} \\ = & \frac{1}{4} \end{array}$

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