Evaluate the limit Lim (x²+x)/(sin(x))x→0
To evaluate the limit of (x^2 + x) / sin(x) as x approaches 0, we can use L’Hôpital’s rule if the limit is in an indeterminate form of 0/0 or ∞/∞
To evaluate the limit of (x^2 + x) / sin(x) as x approaches 0, we can use L’Hôpital’s rule if the limit is in an indeterminate form of 0/0 or ∞/∞.
Applying L’Hôpital’s rule, we take the derivative of the numerator and the denominator separately:
d/dx (x^2 + x) = 2x + 1
d/dx (sin(x)) = cos(x)
Now, we can re-evaluate the limit:
Lim (x^2 + x) / sin(x) as x → 0 = Lim (2x + 1) / cos(x) as x → 0
Now plugging in x = 0, we get:
Lim (2(0) + 1) / cos(0) = Lim 1 / cos(0) as x → 0
The value of cos(0) is 1, so we have:
Lim 1 / 1 = 1 as x → 0
Therefore, the limit of (x^2 + x) / sin(x) as x approaches 0 is equal to 1.
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