Use L'Hôpital's Rule to evaluate the following limit:
lim(x -> 0) (sin(x) - x) / (1 - cos(x))
Answer:
To use L'Hôpital's Rule, we need to find the derivative of the numerator and the derivative of the denominator.
Derivative of numerator:
f'(x) = d/dx (sin(x) - x)
= cos(x) - 1
Derivative of denominator:
g'(x) = d/dx (1 - cos(x))
= sin(x)
Now we can rewrite the limit using the derivatives:
lim(x -> 0) (sin(x) - x) / (1 - cos(x)) = lim(x -> 0) (cos(x) - 1) / sin(x)
Again, we can apply L'Hôpital's Rule to the new limit by finding the derivatives of the new numerator and denominator.
Derivative of new numerator:
f''(x) = d/dx (cos(x) - 1)
= -sin(x)
Derivative of new denominator:
g''(x) = d/dx (sin(x))
= cos(x)
Now we can rewrite the limit once again:
lim(x -> 0) (cos(x) - 1) / sin(x) = lim(x -> 0) -sin(x) / cos(x)
Since we still have an indeterminate form (0/0), we can apply L'Hôpital's Rule one more time.
Derivative of the new numerator:
f'''(x) = d/dx (-sin(x))
= -cos(x)
Derivative of the new denominator:
g'''(x) = d/dx (cos(x))
= -sin(x)
Finally, we can rewrite the limit one last time:
lim(x -> 0) -sin(x) / cos(x) = lim(x -> 0) -cos(x) / -sin(x) = lim(x -> 0) cos(x) / sin(x)
Now let's evaluate the limit using the derived expression:
lim(x -> 0) cos(x) / sin(x) = cos(0) / sin(0) = 1 / 0
At this point, we have an indeterminate form (∞/∞). To resolve this, we need to consider the behavior of the trigonometric functions near 0.
As x approaches 0 from the left side (x < 0), both cos(x) and sin(x) become positive values. Therefore, the limit approaches positive infinity.
As x approaches 0 from the right side (x > 0), both cos(x) and sin(x) become negative values. Therefore, the limit approaches negative infinity.
Since the limit approaches different values from the left and right sides, the overall limit does not exist.
Final Answer: The limit lim(x -> 0) (sin(x) - x) / (1 - cos(x)) does not exist.