Question

Can someone please explain this question?

Answer 1

Answer:

The limit does not exist

Step-by-step explanation:

lim as we approach from below 7

x+5 when x=7  

7+5

12

lim as we approach from above 7

5x-25 when x=7  

5*7 -25

35-25

10

The limit does not exist since these numbers are not the same

Answer 2

Answer:

Limit $7^{-}$ (left hand side limit) is 12

Limit $7^{+}$ (right hand side limit) is 10

Limit at 7 DOES NOT EXIST

Step-by-step explanation:

• Limit $7^{-}$ means the limit from left side of 7
• Limit $7^{+}$ means the limit from left side of 7
• Limit at 7 will exist and would be equal IF BOTH SIDE limits are EQUAL, or else LIMIT DOESN'T EXIST!

Limit $7^{-}$:

When we want to find limit of left side of 7, we take the first part of the function shown (since its defined for $x\leq7$).

We check values very close to 7 but less than it:

• checking 6.98: $6.98+5=11.98$
• checking 6.99: $6.99+5=11.99$

So the limit is approaching 12.

Thus, left hand side limit is 12.

Limit $7^{+}$:

When we want to find limit of right side of 7, we take the second part of the function shown (since its defined for $x>7$).

We check values very close to 7 but greater than it:

• checking 7.02: $5(7.02)-25=10.1$
• checking 7.01: $5(7.01)-25=10.05$

So the limit is approaching 10.

Thus, right hand side limit is 10.

Limit at 7:

If both right hand side limit and left hand side limit equal, then the limit at that specific number exists and is equal to it.

Since we see that the left hand limit (12) and the right hand limit (10) ARE NOT EQUAL, so we conclude that the limit at 7 for the function given DOES NOT EXIST.