Step by step
The question
Question: find the limit as x → 1 of (x² + 5)/(3x + 4).
lim x → 1 x2 + 53x + 4 =
1 Put the number in
Put 1 in place of x.
The top is 6, and the bottom is 7.
The bottom is not 0, so there is no problem here.
top: 12 + 5 = 6
bottom: 3(1) + 4 = 7
Careful!
L'Hôpital's rule is only for 0/0, or infinity over infinity.
Here it would give 2/3, which is wrong.
Careful: not L'Hôpital: f′g′ → 2 over 3 ✗
So the limit is just the value: 6/7.
lim x → 1 x2 + 53x + 4 = 67 = 67
2 Check
Let's check with a calculator, with x very close to 1.
x = 0.99 gives about 0.858.
x = 0.99: x2 + 53x + 4 ≈ 0.858
And x = 1.01 gives about 0.8563.
Both are close to 6/7.
Right!
x = 1.01: x2 + 53x + 4 ≈ 0.8563
Let's look back.
First: put the number in.
Last: check.
Answer
lim x → 1 x2 + 53x + 4 = 67 = 67