1.9 - Connecting Multiple Representations of Limits

Let g be a function that is decreasing for x<0 and decreasing for x>0. If limx→0g(x)=3, which of the following could represent the function g ?

I and III only Since limx→0−x2−3xx=limx→0−x(x−3)x=limx→0(−x+3)=0+3=3, the expression in I could represent the function g. The graph in III could also represent the function g, since the limit determined by the graph as x approaches 0 is equal to 3. The table in II could not represent the function g, since the values in the table suggest that the function grows without bound as x approaches 0 from both sides, and thus the limit would not exist at x=0.

Let g be a function that is increasing for x<1 and increasing for x>1. If limx→1g(x)=5, which of the following could represent the function g ?

I and III only Since limx→1x2+3x−4x−1=limx→1(x−1)(x+4)x−1=limx→1(x+4)=5, the expression in (I) could represent the function g. The graph in (III) could also represent the function g, since the limit determined by the graph as x approaches 1 is equal to 5. This is the graph of the function in (I). The table in (II) could not represent the function g, since the values in the table suggest that the function grows without bound approaching x=1 from both sides, and thus the limit would not exist at x=1.

If h is a piecewise linear function such that limx→3h(x) does not exist, which of the following could be representative of the function h ?

II and III only Given the data in the table for x<3, it appears that limx→3−h(x)=2. The data in the table for x>3 suggest that limx→3+h(x)=0. Since the right-hand and left-hand limits are not equal, the limit at x=3 does not appear to exist for the function with selected values in this table. The graph in III could also represent the function h, since the graph shows a jump discontinuity at x=3, where the left- and right-hand limits are unequal. The values in the table in II are the same as the values on this graph. The expression in I could not represent the function h, since limx→3−h(x)=12−x=12−3=9 and limx→3+h(x)=4(3)−3=12−3=9, which implies that the limit of the expression in I exists at x=3.

If h is a piecewise linear function such that limx→3h(x) does not exist, which of the following could represent the function h ?

II and III only The table in (II) could represent the function h. The data in the table for x<3 are linear with slope 2, so the limit as x approaches 3 from the left would be 7. The data in the table for x>3 are also linear with slope −1, so the limit as x approaches 3 from the right would be 4. Since the left- and right-hand limits are not equal, the limit at x=3 would not exist for the function represented by this table. The graph in (III) could also represent the function h, since the graph shows a jump discontinuity at x=3, where the left- and right-hand limits are unequal. The values in the table in (II) are the same as the values on this graph. The expression in (I) could not represent the function h, since limx→3−(2x+1)=7 and limx→3+(10−x)=7 implies that the limit of the expression in (I) exists at x=3.

The graph of the function f is shown above. Which of the following could be a table of values for f ?

xf(x)0.954.16240.994.0325171.0013.99681.053.8376 The graph of f shows that limx→1f(x)=4 and f(1)=7. This table is consistent with that information.

The graph of the function f is shown above. Which of the following could be a table of values for f ?

xf(x)−1.053.074−1.0013.001−12−0.9992.998−0.952.924 The graph of ff shows that limx→−1f(x)=3limx→−1f(x)=3 and f(−1)=2f(−1)=2. This table is consistent with that information.