2024/G2/2 - Revision history

Aroldog69360@students.laalliance.org at 17:45, 19 April 2023

2023-04-19T17:45:54Z

← Older revision		Revision as of 17:45, 19 April 2023
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	==2.5 CONTINUITY ==		==2.5 CONTINUITY ==

			- A fraction is <u>continuous</u> at a number "a" if the <math> \lim_{x\to a} f(x) = f(a) </math><br>

			1. <math> \lim_{x\to a} f(x) </math> Exist <br>
			2. f(a) Exist <br>
			Ex. 2 <br>

			a) <math> f(x) = {\frac{x^(2)-x-2}{x-2}} x=2 </math><br>

			b) <math> f(x) = \begin{cases}
			{\frac{1}{x^2}}, & \text{if }x\text{ is ≠ 0} \\
			1, & \text{if }x\text{ is = 0}
			\end{cases} </math><br>

			c) <math> f(x) = \begin{cases}
			{\frac{x^(2)-x-2}{x-2}}, & \text{if }x\text{ is ≠ 2} \\
			1, & \text{if }x\text{ is = 2}
			\end{cases} </math><br>
			<math> \lim_{x\to 2} {\frac{x^(2)-x-2}{x-2}} = \lim_{x\to 2} {\frac{(x-2)(x+1)}{(x-2)}} = x+1 = 3 </math><br>
			<math> = (2)+1=3 </math><br>




	==2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES ==		==2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES ==

Aroldog69360@students.laalliance.org at 17:08, 19 April 2023

2023-04-19T17:08:06Z

← Older revision		Revision as of 17:08, 19 April 2023
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	==2.2 THE LIMIT OF A FUNCTION ==		==2.2 THE LIMIT OF A FUNCTION ==
	Notes go here for 2.2... example:<br>		Notes go here for 2.2... example:<br>
	<math>\lim_{z\to ~~z_0~~} f(z)=f(~~z_0~~)</math>		<math>\lim_{z\to z} [f(x) g(x]) = c \cdot[ \lim_{x\to a} f(x) ] </math>
			===Limits are ALWAYS near the number, NEVER on the number.===

	==2.3 CALCULATING LIMITS USING THE LIMIT LAWS ==		==2.3 CALCULATING LIMITS USING THE LIMIT LAWS ==
			<math> \lim_{x\to a} = c \cdot [\lim_{x\to a} f(x)] </math><br>

	==2.5 CONTINUITY ==		==2.5 CONTINUITY ==

	==2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES ==		==2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES ==
			Horizontal Asymptote or H.A
			<br>
			<math> f(x)=\lim_{x \to \infty}\frac{x^2-1}{x^2+1} =\lim_{x \to \infty}\frac{x^2-1}{x^2+1}\cdot \frac{\frac{1}{x^2}}{\frac{1}{x^2}}=\lim_{x \to \infty}\frac{1-\frac{1}{x^2}}{1+\frac{1}{x^2}}=\frac{1-0}{1+0}=1 </math><br>
			This ultimately means that the equation will APPROACH TO ONE it will not ever be on one.
			<br>
			<math>\lim_{x \to \infty}\frac{1}{x}=0 \lim_{x \to -\infty}\frac{1}{x}=0 </math><br>
			<math>\lim_{x \to \infty}\frac{3x^2-x-2}{5x^2+4x+1}=\lim_{x \to \infty}(\frac{3x^2-x-2}{5x^2+4x+1})\cdot\frac{\frac{1}{x^2}}{\frac{1}{x^2}}=\lim_{x \to \infty}\frac{3-\frac{1}{x}-\frac{2}{x^2}}{5+\frac{4}{x}+\frac{1}{x^2}}=\frac{3-(0)-(0)}{5+(0)+(0)}=\frac{3}{5}</math><br>
			V.A or a vertical asymptote is when the function becomes infinity such as
			<math>\lim_{x \to c}f(x)=\infty</math><br>

	==2.7 DERIVATIVES AND RATES OF CHANGE ==		==2.7 DERIVATIVES AND RATES OF CHANGE ==
			To find the Tangent Line we use <math> \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}
			</math><br> We later apply the points on which we want to find the slope.
			<math>f(x)=x^2 f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}= \lim_{h\to 0}\frac{(x+h)^2-x^2}{h}= \lim_{h \to 0}\frac{x^2+2xh+h^2-x^2)}{h}=\lim_{h\to 0}\frac{h^2+2xh}{h}= \lim_{h\to 0}\frac{(\cancel{h})(h+2x)}{\cancel{h}} </math><br> <math>= \lim_{h \to 0}h+2x=(0)+2x </math><br>

	==2.8 THE DERIVATIVE AS A FUNCTION ==		==2.8 THE DERIVATIVE AS A FUNCTION ==
			<math>{\frac{d}{dx}} [c] = 0 </math> <br>
			<math>{\frac{d}{dx}} [c\cdot f(x)] = c\cdot{\frac{d}{dx}} [f(x)] </math> <br>

			<math>{\frac{d}{dx}} [f(x)\pm g(x)] = {\frac{d}{dx}} [f(x)] \pm {\frac{d}{dx}} [g(x)] </math> <br>
			<math> {\frac{d}{dx}} [x^n] = n \cdot x^{(n-1)} </math> <br>
			<math>{\frac{d}{dx}} [a^x] = \ln(a)a^x </math><br>
			<math> {\frac{d}{dx}} [e^x] = e^x </math><br>

Dvaezazizi@laalliance.org: Created page with "==2.2 THE LIMIT OF A FUNCTION == Notes go here for 2.2... example:
$\lim_{z\to z_0} f(z)=f(z_0)$ ==2.3 CALCULATING LIMITS USING THE LIMIT LAWS == ==2.5 CONTINUITY == ==2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES == ==2.7 DERIVATIVES AND RATES OF CHANGE == ==2.8 THE DERIVATIVE AS A FUNCTION =="

2023-03-28T19:02:04Z

Created page with "==2.2 THE LIMIT OF A FUNCTION == Notes go here for 2.2... example:<br> <math>\lim_{z\to z_0} f(z)=f(z_0)</math> ==2.3 CALCULATING LIMITS USING THE LIMIT LAWS == ==2.5 CONTINUITY == ==2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES == ==2.7 DERIVATIVES AND RATES OF CHANGE == ==2.8 THE DERIVATIVE AS A FUNCTION =="

New page

==2.2 THE LIMIT OF A FUNCTION ==
Notes go here for 2.2... example:<br>
<math>\lim_{z\to z_0} f(z)=f(z_0)</math>

==2.3 CALCULATING LIMITS USING THE LIMIT LAWS ==

==2.5 CONTINUITY ==

==2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES ==

==2.7 DERIVATIVES AND RATES OF CHANGE ==

==2.8 THE DERIVATIVE AS A FUNCTION ==