## Calculus: Early Transcendentals 8th Edition

The equation of the tangent line $l$ at point $A(2,3)$ is $$(l):y=9x-15$$
$$y=f(x)=x^3-3x+1$$ Given point $A(2,3)$ According to definition, the slope of the tangent line $l$ at point $A(2, 3)$ is $$m_l=\lim\limits_{x\to2}\frac{f(x)-f(2)}{x-2}$$$$m_l=\lim\limits_{x\to2}\frac{(x^3-3x+1)-3}{x-2}$$$$m_l=\lim\limits_{x\to2}\frac{x^3-3x-2}{x-2}$$$$m_l=\lim\limits_{x\to2}\frac{(x-2)(x+1)^2}{x-2}$$$$m_l=\lim\limits_{x\to2}(x+1)^2$$$$m_l=(2+1)^2=9$$ Therefore, the equation of the tangent line $l$ would have the following form: $$(l): y=9x+b$$ Since $l$ passes through point $A(2,3)$, we have $$9\times2+b=3$$$$18+b=3$$$$b=-15$$ In conclusion, the equation of the tangent line $l$ at point $A(2,3)$ is $$(l):y=9x-15$$