Topic 5.1: Introduction to Differentiation Questions

If lim_x→1^-f(x)=2 and lim_x→1^+f(x)=2, then lim_x→1f(x) =

Evaluate lim_x→4(x²-16)/(x-4).

If lim_x→2 f(x)=7, which statement is true?

Evaluate lim_x→5(3x-4).

Evaluate lim_x→3(x²-9)/(x-3).

Evaluate lim_x→2(x²+3x-1).

If the left and right limits at x = 4 are different, what can you conclude?

A piecewise function has both one-sided limits equal to 4 at x=a, but f(a)=-3. Determine lim_{x→ a}f(x) and comment on continuity.

A table shows f(x) → 7 as x → 2 from both side. Estimate lim_x→2f(x) and state what the table tells you about f(2).

Evaluate lim_x→0((x+1)²-1)(x).

A graph approaches y=2 as x approaches 1 from both side, but f(1)=10. Find lim_x→1f(x).

For f(x) = (x² − 1)/(x − 1):

Given lim_x→3^-f(x)=5 and lim_x→3^+f(x)=7, explain whether lim_x→3f(x) exist.

A ball is thrown upward and its height after t seconds is h(t) = 20t − 5t² metres.

(a) Find the average rate of change of height between t = 1 and t = 2.

(b) Find the instantaneous rate of change at t = 1 (use h′(t) = 20 − 10t) and compare.

For g(x)=(|x|)/(x), find both one-sided limits at x=0 and state whether the two-sided limit exist.

A population model gives P(t)=(t²-25)/(t-5). Use limits to find the rate indicator at t=5 and interpret your answer.