Find the indicated limits. If the limit does not exist, so state, or use the symbol + ∞ or - ∞.

Question

f(x) = { 2 - x if x ≤ 3
{ -1 + 3x - x^2 if x > 3

a) lim 3+ f(x)
x->3

b) lim 3- f(x)
x->3

c) lim f(x)
x->3

d) lim ∞ f(x)
x->3

e) lim -∞ f(x)
x->3

Answer 1

a) lim 3+ f(x)
x->3

To find the limit as x approaches 3 from the right, we look at the function for x > 3: -1 + 3x - x^2.

So, lim 3+ f(x) = -1 + 3(3) - (3)^2 = -1 + 9 - 9 = -1.

b) lim 3- f(x)
x->3

To find the limit as x approaches 3 from the left, we look at the function for x ≤ 3: 2 - x.

So, lim 3- f(x) = 2 - 3 = -1.

c) lim f(x)
x->3

Since the limits from both the left and the right are equal, the limit at x = 3 exists and is equal to -1.

d) lim ∞ f(x)
x->∞

As x goes to positive infinity, the second part of the function becomes dominant: -1 + 3x - x^2. Since the leading term is x^2 with a negative coefficient, the function will go to negative infinity as x approaches positive infinity:

lim ∞ f(x) = -∞

e) lim -∞ f(x)
x->-∞

As x goes to negative infinity, the first part of the function becomes dominant: 2 - x. Since the coefficient of x is negative, the function will go to positive infinity as x approaches negative infinity:

lim -∞ f(x) = ∞