Answer:
the last option:
Step-by-step explanation:
Make sure you have the numerical answer for each of the functional expressions that are shown among the possible solution choices:
f(4) = -14 (what the blue function reads [its y-value] when x is 4)
g(4) = 10 (what the red function reads [its y-value] for x=4)
g(-2) = 4 (y-value of the red function when x is -2)
f(2) = -8 (y-value of the blue function for x = 2)
f(-2) = 4 (y-value of the blue function for x =-2)
use them to compare the options they give you, and the only one that matches is the last option.
Observe the given figure.
Given:
To prove:
Statement
1.
Reason: Given
2.
Reason: Substitution
3. are alternate interior angles.
Reason: Definition of alternate interior angles
4.
Reason: If alternate interior angles are equal, then the lines are parallel.
means to say that for any given , we can find
such that anytime
(i.e. the whenever
is "close enough" to 5), we can guarantee that
(i.e. the value of
is "close enough" to the limit value).
What we want to end up with is
Dividing both sides by 3 gives
which suggests is a sufficient threshold.
The proof itself is essentially the reverse of this analysis: Let be given. Then if
and so the limit is 7. QED