Problem to solve
Compute \[ \lim_{ x \to 2 } \frac{\sqrt{x^2+5}-3}{x^2-2x} \]
Solution
STEP 1
Find the following limit:
lim_(x->2) (sqrt(x^2 + 5) - 3)/(x^2 - 2 x)
STEP 2
Hint: | Rationalize the expression.
(sqrt(x^2 + 5) - 3)/(x^2 - 2 x) = ((sqrt(x^2 + 5) - 3) (sqrt(x^2 + 5) + 3))/((x^2 - 2 x) (sqrt(x^2 + 5) + 3)) = (x + 2)/(x (sqrt(x^2 + 5) + 3)):
lim_(x->2)((x + 2)/(x (sqrt(x^2 + 5) + 3)))
STEP 3
Hint: | The limit of a continuous function at a point is just its value there.
lim_(x->2) (x + 2)/(x (sqrt(x^2 + 5) + 3)) = (2 + 2)/(2 (sqrt(2^2 + 5) + 3)) = 1/3:
Answer: | 
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