2024/G2/2

2.2 THE LIMIT OF A FUNCTION

Notes go here for 2.2... example:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{z\to z} [f(x) g(x]) = c \cdot[ \lim_{x\to a} f(x) ] }

2.3 CALCULATING LIMITS USING THE LIMIT LAWS

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a} = c \cdot [\lim_{x\to a} f(x)] }

2.5 CONTINUITY

- A fraction is continuous at a number "a" if the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a} f(x) = f(a) }

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a} f(x) } Exist
2. f(a) Exist
Ex. 2

a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = {\frac{x^(2)-x-2}{x-2}} x=2 }

b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} {\frac{1}{x^2}}, & \text{if }x\text{ is ≠ 0} \\ 1, & \text{if }x\text{ is = 0} \end{cases} }

c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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} }

         Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  \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 }


                                      Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  = (2)+1=3 }

2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES

Horizontal Asymptote or H.A
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 }
This ultimately means that the equation will APPROACH TO ONE it will not ever be on one.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to \infty}\frac{1}{x}=0 \lim_{x \to -\infty}\frac{1}{x}=0 }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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}}
V.A or a vertical asymptote is when the function becomes infinity such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c}f(x)=\infty}

2.7 DERIVATIVES AND RATES OF CHANGE

To find the Tangent Line we use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} }
We later apply the points on which we want to find the slope. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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}} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle = \lim_{h \to 0}h+2x=(0)+2x }

2.8 THE DERIVATIVE AS A FUNCTION

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [c] = 0 }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [c\cdot f(x)] = c\cdot{\frac{d}{dx}} [f(x)] }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [f(x)\pm g(x)] = {\frac{d}{dx}} [f(x)] \pm {\frac{d}{dx}} [g(x)] }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [x^n] = n \cdot x^{(n-1)} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [a^x] = \ln(a)a^x }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [e^x] = e^x }