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Lesson Example Discussion Quiz: Class Homework

Step-1

Title: Limits and continuity

Grade: 1400-a Lesson: S2-L4

Explanation: Hello Students, time to practice and review the steps for the problem.

Lesson Steps

Step	Type	Explanation	Answer
1	Problem	Evaluate the following : \$\lim_{x \to 0} (\sqrt(2 + x) - \sqrt2)/x \$.
2	Step	Given that	\$\lim_{x \to 0} {(\sqrt(2 + x) - \sqrt2)/x} \$
3	Step	Rationalize the numerator: Multiply both the numerator and the denominator by the conjugate of the numerator	\$ (\sqrt(2 + x) - \sqrt2)/x times (\sqrt(2 + x) + \sqrt2)/ (\sqrt(2 + x) + \sqrt2) \$ \$= ( (\sqrt(2 + x) - \sqrt2)(\sqrt(2 + x) + \sqrt2))/(x times (\sqrt(2 + x) + \sqrt2)) \$
4	Formula:	Now applying \$ a^2 - b^2 = (a + b) (a - b) \$	\$= (\sqrt(2 + x)^2 - \sqrt(2)^2)/(x times (\sqrt(2 + x) + \sqrt2)) \$ \$= 1/(\sqrt(2 + x) + \sqrt2) \$
5	Step	Let’s simplify the expression by evaluating the limit as x approaches 0	\$\lim_{x \to 0} {1/(\sqrt(2 + x) + \sqrt2)} \$ \$ = 1/(\sqrt(2 + 0) + \sqrt2) \$ \$ = 1/(2\sqrt2) \$
6	Step	Thus, the evaluation of the given limit is \$ 1/(2\sqrt2) \$.
7	Choice.A	This option suggests that the limit is \$1/\sqrt2\$. However, our evaluation yielded \$1/(2\sqrt2)\$, which is not equal to \$1/\sqrt2\$. So, option A is incorrect	\$ 1/(\sqrt2) \$
8	Choice.B	This option suggests that the limit is \$\sqrt2\$. However, our evaluation yielded \$1/(2\sqrt2)\$, which is not equal to \$\sqrt2\$. So, option B is incorrect	\$ \sqrt2 \$
9	Choice.C	This option suggests that the limit is \$ (2)/(\sqrt2) \$. However, our evaluation yielded \$1/(2\sqrt2)\$, which is not equal to \$ (2)/(\sqrt2) \$. So, option C is incorrect	\$ (2)/(\sqrt2) \$
10	Choice.D	This option is the same as our evaluated result. So, if our evaluation is correct, this option is correct	\$ 1/(2\sqrt2) \$
11	Answer	Option	D
12	Sumup	Can you summarize what you’ve understood in the above steps?

Step

Type

Explanation

Answer

Problem

Evaluate the following : \$\lim_{x \to 0} (\sqrt(2 + x) - \sqrt2)/x \$.

Step

Given that

\$\lim_{x \to 0} {(\sqrt(2 + x) - \sqrt2)/x} \$

Step

Rationalize the numerator: Multiply both the numerator and the denominator by the conjugate of the numerator

\$ (\sqrt(2 + x) - \sqrt2)/x times (\sqrt(2 + x) + \sqrt2)/ (\sqrt(2 + x) + \sqrt2) \$

\$= ( (\sqrt(2 + x) - \sqrt2)(\sqrt(2 + x) + \sqrt2))/(x times (\sqrt(2 + x) + \sqrt2)) \$

Formula:

Now applying \$ a^2 - b^2 = (a + b) (a - b) \$

\$= (\sqrt(2 + x)^2 - \sqrt(2)^2)/(x times (\sqrt(2 + x) + \sqrt2)) \$
\$= 1/(\sqrt(2 + x) + \sqrt2) \$

Step

Let’s simplify the expression by evaluating the limit as x approaches 0

\$\lim_{x \to 0} {1/(\sqrt(2 + x) + \sqrt2)} \$
\$ = 1/(\sqrt(2 + 0) + \sqrt2) \$
\$ = 1/(2\sqrt2) \$

Step

Thus, the evaluation of the given limit is \$ 1/(2\sqrt2) \$.

Choice.A

This option suggests that the limit is \$1/\sqrt2\$. However, our evaluation yielded \$1/(2\sqrt2)\$, which is not equal to \$1/\sqrt2\$. So, option A is incorrect

\$ 1/(\sqrt2) \$

Choice.B

This option suggests that the limit is \$\sqrt2\$. However, our evaluation yielded \$1/(2\sqrt2)\$, which is not equal to \$\sqrt2\$. So, option B is incorrect

\$ \sqrt2 \$

Choice.C

This option suggests that the limit is \$ (2)/(\sqrt2) \$. However, our evaluation yielded \$1/(2\sqrt2)\$, which is not equal to \$ (2)/(\sqrt2) \$. So, option C is incorrect

\$ (2)/(\sqrt2) \$

Choice.D

This option is the same as our evaluated result. So, if our evaluation is correct, this option is correct

\$ 1/(2\sqrt2) \$

Answer

Option

Sumup

Can you summarize what you’ve understood in the above steps?