Final answer:
To prove the limit as x approaches 1 of (x²+3x+5) is 9, we can evaluate the expression and show that it approaches 9 as x gets arbitrarily close to 1.
Step-by-step explanation:
To prove the limit as x approaches 1 of (x²+3x+5) is 9 using the definition of limits, we will start by evaluating the expression as x gets arbitrarily close to 1. Let's substitute 1 into the expression: (1²+3(1)+5) = 1+3+5 = 9, which is the expected result. This verifies that the expression is equal to 9 at x=1. Now, we need to show that the expression approaches 9 as x approaches 1. To do this, we can use the fact that for any positive number ε, we can always find a positive number δ such that |x-1|<δ implies |(x²+3x+5) - 9|<ε.