In the description of end behaviour on the previous page, the phrase tends towards the x-axis was introduced. Look more closely at what this means by studying the table of values for the exponential functions y = 2x and y = (0.5)x at extreme values of x.

x
y = 2x
−50
 2−50 0.000 000 000 000 000 888 2
−25
 2−25 0.000 000 029 80
−10
 2−10 0.000 976 5
−5
 2−5 = 0.312 5
−1
 2−1 = 0.5
0
 20 = 1
1
 21 = 2
5
 25 = 32
10
 210 = 1 024
25
 225 = 33 554 432
50
 250 = 1 130 000 000 000 000

As the x-value decreases, the y-value gets closer to zero. Because of this, you say the function approaches, or tends towards, y = 0, which is the x-axis.

Note that no real number can be substituted for x in the exponent to make the y-value equal to zero or a negative.

As the x-value increases, the y-value gets extremely large. The y-value will continue to increase, so you say the function approaches positive infinity.
x
y = (0.5)x
−50
 (0.5)−50 1 125 899 907 000 000
−25
 (0.5)−25 = 33 554 432
−10
 (0.5)−10 = 1 024
−5
 (0.5)−5 = 32
−1
 (0.5)−1 = 2
0
 (0.5)0 = 1
1
 (0.5)1 = 0.5
5
 (0.5)5 = 0.312 5
10
 (0.5)10 0.000 976 6
25
 (0.5)25 0.000 000 029 80
50
 (0.5)50 0.000 000 000 000 000 888 2

As the x-value decreases, the y-value gets extremely large. The y-value will continue to increase, so you say the function approaches positive infinity.

As the x-value increases, the y-value gets closer to zero. Because of this, you say the function approaches, or tends towards, y = 0, which is the x-axis.