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Achievement objective M8-10

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • identify discontinuities and limits of functions.


  • Links features of graphs with the limiting behaviour of functions.
  • Uses limiting features of functions to sketch graphs.
  • Finds limits algebraically, graphically, and numerically by considering behaviour as:
    • x approaches a specific value from above and below
    • x tends towards +∞ or -∞
  • Demonstrates understanding of continuity at a point (the limit as x tends to a of f(x) = f(a)).
  • Identifies discontinuities graphically or algebraically.
  • Informally links concepts of continuity and differentiability M7-9.
  • See key mathematical ideas on NZmaths.


M8-10 links from M7-9, M8-2, M8-7.

Possible context elaborations

  Possible context elaboration 1, equation 1.

If you cannot view or read this equation, open this text version.

  A graph of the linear function y equals x plus two.

The limit as x approaches zero of 1 divided by x.  does not exist. This limit does not exist since the limit from the left decreases without bound and the limit from the right increases without bound.

  A graph of the rational function y equals 1  divided by x.

Possible context elaboration 1, equation 3. This means the limit does not exist but that we know about the behaviour of the function as x approaches 3. The function does not approach a number from both sides, but it tends to +∞ (that is, the function increases without bound) from both sides of x = 3.

If you cannot view or read this equation, open this text version.

  Possible context elaboration 1, graph 3.

If you cannot view or read this graph, open this text version.

  • Sketch of f(x) = x , x < 3 and f(x) = x + 2, x ≥ 3 and comment The limit as x approaches 3 of f ( x ).  does not exist and so it is discontinuous at x = 3.
  • Discontinuities shown by a graph with a jump, for example, piecewise function.
  • Discontinuities shown by a graph with a hole, for example, y=(x2-1)/(x-1) which has a hole at x=1.
  • Graf it: Exploring unfamiliar graphs.

Assessment for qualifications

NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The NZQA subject-specific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.

Learn more:

The following achievement standard(s) could assess learning outcomes from this AO:

  • AS91578 Mathematics and statistics 3.6 Apply differentiation methods in solving problems; External, 6 credits.

Refer to the mathematics and statistics matrix.

Last updated September 26, 2013