Problem to solve
Compute \[ \lim_{ x \to 3^+ } \frac{x+2}{x-3} \]
Solution
STEP 1
Find the following limit:
lim_(x->3^+) (x + 2)/(x - 3)
STEP 2
Hint: | If neither of the terms of a product approach 0 as x->a, then by the product rule, the limit of the product is the product of the limits.
Applying the product rule, write lim_(x->3^+) (x + 2)/(x - 3) as (lim_(x->3^+) 1/(x - 3)) lim_(x->3^+)(x + 2):
lim_(x->3^+)(x + 2) lim_(x->3^+) 1/(x - 3)
STEP 3
Hint: | The limit of a continuous function at a point is just its value there.
lim_(x->3^+)(x + 2) = 3 + 2 = 5:
5 lim_(x->3^+) 1/(x - 3)
STEP 4
Hint: | If lim_(x->3^+) y = 0, then the limit of 1/y is ± ∞.
Since lim_(x->3^+)(x - 3) = 0 and x - 3>0 for all x just to the right of x = 3, lim_(x->3^+) 1/(x - 3) = ∞:
5 ∞
STEP 5
Hint: | Evaluate 5 ∞.
5 ∞ = ∞:
Answer: | 
 | ∞