Thinking about Functions
Building blocks

Approaching asymptotes
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Descriptions and graphs

How would you describe what an asymptote is?

Here are some descriptions or statements about asymptotes.

  1. “An asymptote is a line which a curve gets closer and closer to but doesn’t meet.”

  2. “An asymptote is a line which a curve approaches as \(x\) tends to infinity.”

  3. “A curve can’t cross an asymptote.”

  4. “Asymptotes are parallel to the coordinate axes.”

  5. “A graph can only have one asymptote parallel to each axis.”

  6. “Asymptotes occur when a function isn’t defined for certain input values.”

  7. “A function tends to positive infinity on one side of an asymptote and tends to negative infinity on the other side.”

Here are some examples of curves, some of which have asymptotes. Use these to help you decide whether you agree with the statements above.

Downward sloping curve tending to a horizontal line as x tends to positive infinity.
Graph of cube root x; an upward sloping curve tangent to and crossing the y-axis at the origin.
Curve which crosses the x-axis but also tends to the x-axis as x tends to positive infinity.
This curve has two asymptotes, one positively sloped and one negatively sloped both passing through the origin together forming a cross shape. The two components of the curve lie in the left and right sections of this cross shape.
Curve of the logarithm function; an always increasing function with always decreasing slope which tends to negative infinity as x tends to 0 from the right.
A curve which has a maximum at x=0 and tends to the x-axis from above as x tends to both positive and negative infinitity.
Graph of the parabola x squared
Curve which is a curvy v shape with vertex at the origin
Graph of square root x, translated to the right
Oscillating curve with decreasing amplitude that tends to 0 as x tends to positive infinity.
Graph of tanh x; a curve tending to a horizontal line below the x-axis as x tends to negative infinity, passing through the origin, and tending to a horizontal line above the x-axis as x tends to positive infinity.
Curve consisting of two components one in the second quadrant and one in the fourth. The curve tends to a negative sloping line as x tends to both positive and negative infinity and the curve tends to positive infinity and negative infinity as x tends to 0 from the left and right respectively.
Graph of tan x
Curve passing through the origin with positive slope and tending to a horizontal line above the x-axis as x tends to positive infinity.
This curve has 3 components; one component is like the component of the graph of tan x which passes throught the origin, the other two components tend towards the line which is tangent to tan x at the origin, from above as x tends to negative infinity and from below as x tends to positive infinity. These components also tend towards vertical lines as they approach the tan x component
This curve has the shape of an upside down funnel; it tends towards a horizontal line from above as x tends to both positive and negative infinity, and at a particular value of x it tends to positive infinity from both the left and the right.
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Last updated 10-Nov-16
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