Topic 5.1: Introduction to Differentiation Questions

Evaluate lim_(x→2)(x²+3x-1).

Evaluate lim_(x→4)(x²-16)/(x-4).

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

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).

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

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→2)f(x) and state what the table tells you about f(2).

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→3)f(x) exist.

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

Evaluate lim_(x→0)((x+1)²-1)(x).

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.

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

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