Explain why $\underset{x1}{lim}\frac{1}{x-1}$ does not exist. $[]$

Consider the graph of the function $f(x)=5x^2-8x.$ Calculate the slope of the secant that joins the points on the graph given by $x=-2$ and $x=1.[A=2$

$3.$For the function shown below, determine the following: $]=[3]=[3$

a$)\underset{x1}{lim}f(x)$

b$)\underset{x2}{lim}f(x)$

c$)\underset{x{4}^{-}}{lim}f(x)$

d$)$ values of $x$ for which $f(x)$ is discontinuous

A weather balloon is rising vertically. After $t$ hours, its distance above the ground, measured in kilometres, is given by the formula $]=[4]=[2$

a$)$ Calculate the average velocity of the weather balloon from $t=2h$ to $t=5h.$

b$)$ Determine its instantaneous velocity at $t=3h.$

Calculate and solve the average rate of change in $f(x)=\sqrt[\phantom{2}]{x+11}$ with respect to $x$ from $x=5$ to $x=5+h.$ Determine the instantaneous rate of change at $]=[2]=[2$

Determine the slope of the tangent at $x=4$ for $]=[2]=[2$

Evaluate the following limits & Explain the answers: $]=[6]=[6]=[6$

a$)\underset{x3}{lim}\frac{4x^2-36}{2x-6}$

b$)\underset{x2}{lim}\frac{2x^2-x-6}{3x^2-7x+2}$

c$)\underset{x5}{lim}\frac{x-5}{\sqrt[\phantom{2}]{x-1}-2}$

d$)\underset{x-1}{lim}\frac{x^3+1}{x^4-1}$

e$)\underset{x3}{lim}(\frac{1}{x-3}-\frac{6}{x^2-9})$

f$)\underset{x0}{lim}\frac{(x+8{)}^{\frac{1}{8}}-2}{x}$

Solve constants a and $b$ such that $f(x)$ is continuous for all values of $x.[A=4$

$f(x)=\{\begin{array}{ccc}{x}^2+bx+a,& if& x<mn>5</mn>\\ 8,& if& x=5\\ ax+3,& if& x>5\end{array}$