Answered: lim (x2 +x³) = 0 x+0

Math

Calculus

lim (x2 +x³) = 0 x+0

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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Concept explainers

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.

Question

Use the formal definition of the limit to prove the statement rigorously.

Expert Solution

Step 1

To prove: $\lim_{x \to 0} (x^{2} + x^{3}) = 0$ .

We know that $\lim_{x \to a} [f (x) + g (x)] = \lim_{x \to a} f (x) + \lim_{x \to a} g (x)$

Therefore,

$\begin{array}{rcl} \lim_{x \to 0} (x^{2} + x^{3}) & = & \lim_{x \to 0} x^{2} + \lim_{x \to 0} x^{3} \\ = & 0^{2} + 0^{3} \\ = & 0 + 0 \\ = & 0 \end{array}$

Therefore, LHS=RHS

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$Limits and Continuity$

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