2024/G8/3

Section 1
Section 2

3.1 THE LIMIT OF 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 \mathbf{New rules}}

Review section 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}} [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}} [a^x] = \ln(a)a^x }

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$\color {green}Power\,Rule$

${\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1$

3.2 CALCULATING LIMITS USING THE LIMIT LAWS

$\mathbf {New\ rules}$

$\color {Blue}Product\,Rule$

${\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}$

$\color {Red}Quotient\,Rule$

${\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}$

$\mathbf {Examples}$

$\mathbf {Ex.1}$

If $f(x)=x\cdot {e^{x}}$

$f^{\prime }(x)=1\cdot {e^{x}}+x\cdot {e^{x}}$
$f^{\prime }(x)=e^{x}(1+x)$

$\mathbf {Ex.2}$

If $f(t)={\sqrt {t}}(a+bt)$

$f^{\prime }(t)={\frac {1}{2{\sqrt {t}}}}(a+bt)+t{\sqrt {t}}(b)$

$\mathbf {Ex.3}$

If $f(x)={\sqrt {x}}\cdot {g(x)}$

$g(4)=2$

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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^\prime(x)=\frac{1}{2\sqrt{x}}\cdot{g(x)}+\sqrt{x}\cdot{g^\prime(x)}}
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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^\prime(x)=6.5}

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If 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 y=\frac{x^2+x-2}{x^3+6}}
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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 y^\prime=\frac{-x^4-2x^3+6x+12x+6}{(x^3+6)^2}}