10. + ∞
Here we have to think about what happens when we plug in a value that is very close to 7, but
a little bit more. The top expression will approach 7. The bottom expression will approach 0,
but will be a little bit positive. Thus, the limit will be , which is +∞.
- The limit Does Not Exist.
In order to evaluate the limit as x approaches 7, we find the limit as it approaches 7+ (from the
right) and the limit as it approaches 7− (from the left). If the two limits approach the same
value, or both approach positive infinity or both approach negative infinity, then the limit is that
value, or the appropriately signed infinity. If the two limits do not agree, the limit “Does Not
Exist.” Here, if we look at the solutions to problem 14, we see that as x approaches 7+, the
limit is +∞. As x approaches 7−, the top expression will approach 7. The bottom will approach
0, but will be a little bit negative. Thus, the limit will be , which is −∞. Because the two
limits are not the same, the limit Does Not Exist.
- (a) 4; (b) 5; (c) The limit Does Not Exist.
(a) Notice that f(x) is a piecewise function, which means that we use the function f(x) = x^2 − 5
for all values of x less than or equal to 3. Thus, f(x) = 3^2 − 5 = 4.
(b) Here we use the function f(x) + 2 for all values of x greater than 3. Thus, f(x) = 3 + 2 =
5.
(c) In order to evaluate the limit as x approaches 3, we find the limit as it approaches 3+ (from
the right) and the limit as it approaches 3− (from the left). If the two limits approach the same
value, or both approach positive infinity or both approach negative infinity, then the limit is that
value, or the appropriately signed infinity. If the two limits do not agree, the limit “Does Not
Exist.” Here, if we refer to the solutions in parts (a) and (b), we see that f(x) = 4 and
f(x) = 5. Because the two limits are not the same, the limit Does Not Exist.
