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 left of x = 3, lim_(x->3^-) 1/(x - 3) = -∞: 5 -∞
STEP 5
Hint: | Evaluate 5 (-∞). 5 (-∞) = -∞: Answer: | | -∞
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Edited by tododemate® - 2021