Question

Calculus I

$lim \frac{x^{4} - 3x^{3} + x^{2} - 3x}{2x-6} \\x-\ \textgreater \ 3$

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\: \displaystyle \lim_{x \: \to \: 3} \:\dfrac{ {x}^{4} - {3x}^{3} + {x}^{2} - 3x}{2x - 6}$

If we substitute x = 3 directly, we get

$\rm \: \: = \: \:\dfrac{ {(3)}^{4} - {3(3)}^{3} + {(3)}^{2} - 3 \times 3}{2 \times 3 - 6}$

$\rm \: \: = \: \dfrac{81 - 81 + 9 - 9}{6 - 6}$

$\rm \: \: = \: \dfrac{0}{0} \: which \: is \: indeterminant \: form$

Consider,

$\rm :\longmapsto\: \displaystyle \lim_{x \: \to \: 3} \:\dfrac{ {x}^{4} - {3x}^{3} + {x}^{2} - 3x}{2x - 6}$

$\rm \: \: = \:\displaystyle \lim_{x \: \to \: 3} \: \dfrac{ {x}^{3}(x - 3) + x(x - 3)}{2(x - 3)}$

$\rm \: \: = \:\displaystyle \lim_{x \: \to \: 3} \: \dfrac{ ({x}^{3} + x) \: \cancel{ (x - 3)}}{2 \: \cancel{(x - 3)}}$

$\rm \: \: = \: \displaystyle \lim_{x \: \to \: 3} \:\dfrac{ {x}^{3} + x}{2}$

$\rm \: \: = \: \dfrac{ {(3)}^{3} + 3 }{2}$

$\rm \: \: = \: \dfrac{27 + 3}{2}$

$\rm \: \: = \: \dfrac{30}{2}$

$\rm \: \: = \: 15$

Hence,

$\bf\implies \:\: \displaystyle \lim_{x \: \to \: 3} \bf \:\dfrac{ {x}^{4} - {3x}^{3} + {x}^{2} - 3x}{2x - 6} = 15$

Additional Information :-

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{sin \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{tan \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{tan^{ - 1} \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(4)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{sin^{ - 1} \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(5)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{log(1 \: + \: x)}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(6)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{ {e}^{x} - 1 }{x} \: = \: 1 }}}}}} \\ \end{gathered}$

Answer 2

x+1=0 x = -1

a(-1)^3 + 3(-1)^2 -13 = -a-10

5(-1)^3 -8(-1) +a = a+3

a+3 = -a-10 ➜ 2a = -13 → a= -13/2 = -6.5