Explain why \lim\limits_{x\to 1} \displaystyle\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.

For the function shown below, determine the following:

(a) \lim\limits_{x\to 1} f(x)

(b) \lim\limits_{x\to 2} f(x)

(c) \lim\limits_{x\to 4^-} f(x)

(d) Values of x for which f is discontinuous

A weather balloon is rising vertically. After t hours, its distance above the ground, measured in kilometres, is given by the formula s(t)=8t-t^2.

Determine the average velocity of the weather balloon from t=2 h to t=5 h.

A weather balloon is rising vertically. After t hours, its distance above the ground, measured in kilometres, is given by the formula s(t)=8t-t^2.

Determine its velocity at t=3 h.

Determine the average rate of change in f(x)=\sqrt{x+11} with respect to x from x=5 to `x=5+h.

Determine the slope of the tangent at x=4 from f(x)=\displaystyle\frac{x}{x^2-15}.

Evaluate the following limits:

\lim\limits_{x\to 3}\displaystyle{\frac{4x^2-36}{2x-6}}

Evaluate the following limits:

\lim\limits_{x\to 2}\displaystyle{\frac{2x^2-x-6}{3x^2-7x+2}}

Evaluate the following limits:

\lim\limits_{x\to 5}\displaystyle{\frac{x-5}{\sqrt{x-1}-2}}

Evaluate the following limits:

\lim\limits_{x\to -1}\displaystyle{\frac{x^3+1}{x^4-1}}

Evaluate the following limits:

\lim\limits_{x\to 3} \displaystyle{\left(\frac{1}{x-3}-\frac{6}{x^2-9}\right)}

Evaluate the following limits:

\lim\limits_{x\to 0}\displaystyle{\frac{(x+8)^{1/3}-2}{x}}

Determine constants a and b such that f(x) is continuous for all values of x.

f(x)= \begin{cases} ax+3, & \text{if } x>5 \\ 8, & \text{if } x=5 \\ x^2+bx+a & \mathrm{if } x<5 \end{cases}