2024/G1/2

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 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\to a}=c\cdot [\lim _{x\to a}f(x)]}

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 }

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 }