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Limits and Introduction to Calculus

Extension Problems

For the function $f\left(x\right)=ax^3+bx+c$, the average rate of change from $x=1$ to $x=2$ is $3$ and the average rate of change from $x=2$ to $x=3$ is $-9$. Find the values of $a$ and $b$.

$a=$

$b=$

Evaluate the following limits. If the limit does not exist, enter “DNE”.

$\displaystyle \lim \limits_{x \to 0}{\frac{1}{x}\left(\frac{1}{1+x}-1\right)}$

$\displaystyle \lim\limits_{x \to 1} \frac{\sqrt{x+3}-2}{x-1}=$

$\displaystyle \lim\limits_{x \to \infty} \frac{2x-1}{\sqrt{x^2+1}}=$

$\displaystyle \lim \limits_{x \to 1^-}{\frac{\vert x-1\vert + \vert 1-x\vert}{1-x}}$

Evaluate the following limits. If the limit does not exist, enter “DNE”.

$\displaystyle \lim\limits_{x \to \infty} \sqrt{x^2-3x+1}-x=$

$\displaystyle \lim\limits_{x \to -\infty} x\left(\sqrt{x^2+2}+x\right)=$

Evaluate $\displaystyle \lim\limits_{x \to 1} \frac{1-\sqrt[3]{x}}{x-1}$. $\;\left[ \text{hint: } a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right) \right]$

If $f\left(x\right)=x^4-2x^3+1$, find $\displaystyle \lim \limits_{x \to 2}{\frac{f\left(x\right)-f\left(2\right)}{x-2}}$

$f(x)$ is differentiable at $x=1$ and $f'(1)=5$. Find the value of $\displaystyle \lim \limits_{x \to 1}{\frac{x^2-1}{f(x)-f(1)}}$.

If $f(x)=x^3+ax^2+bx+c$ and $\left(x-2\right)f'(x)=3f(x)$, find the values of $a, b$ and $c$.

$a=$

$b=$

$c=$