Evaluate the limit.limx→0 x^2/x+6+(36/x−6)
To evaluate the limit as x approaches 0 for the expression (x^2)/(x+6) + (36/(x-6)), we can simplify the expression first and then substitute the limit value
To evaluate the limit as x approaches 0 for the expression (x^2)/(x+6) + (36/(x-6)), we can simplify the expression first and then substitute the limit value.
Step 1: Simplifying the expression
Let’s start by finding the least common denominator (LCD) for the two fractions, which is (x+6)(x-6). Then, we can rewrite the expression with a common denominator:
(x^2/(x+6)) + (36/(x-6)) = [(x^2 * (x-6)) + (36 * (x+6))] / [(x+6)(x-6)]
Next, simplify the numerator:
(x^2 * (x-6)) + (36 * (x+6)) = x^3 – 6x^2 + 36x + 216
Now, rewrite the expression with the simplified numerator and the common denominator:
(x^3 – 6x^2 + 36x + 216) / [(x+6)(x-6)]
Step 2: Evaluating the limit
To evaluate the limit as x approaches 0, we substitute 0 for x in the simplified expression:
lim(x→0) [(x^3 – 6x^2 + 36x + 216) / [(x+6)(x-6)]]
Plugging in x=0 yields:
[(0^3 – 6(0)^2 + 36(0) + 216) / [(0+6)(0-6)]]
Simplifying further:
[216 / (6)(-6)]
Now, evaluating the fraction:
216 / -36 = -6
Therefore, the limit as x approaches 0 for the given expression is -6.
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