Use L'Hôpital's Rule (possibly more than once) to evaluate the following limit. (Use symbolic notation and fractions where needed.) lim_(x rarr0)(sin(7x)-7x cos(7x))/(7x-sin(7x))=

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Use L'Hôpital's Rule (possibly more than once) to evaluate the following limit. (Use symbolic notation and fractions where needed.)  lim_(x rarr0)(sin(7x)-7x cos(7x))/(7x-sin(7x))=

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Check Indeterminate Form: We are given the limit to evaluate using L'Hôpital's Rule: lim_(x -> 0) (sin(7x) - 7x cos(7x)) / (7x - sin(7x)) First, we need to check if the limit is in an indeterminate form that allows us to apply L'Hôpital's Rule.Evaluate Numerator and Denominator: Evaluate the limit of the numerator and the denominator separately as x approaches 0: lim_(x -> 0) sin(7x) = 0 lim_(x -> 0) 7x cos(7x) = 0 lim_(x -> 0) 7x = 0 lim_(x -> 0) sin(7x) = 0 Both the numerator and the denominator approach 0, so we have an indeterminate form of 0/0.Apply L'Hôpital's Rule: Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that if lim_(x -> c) f(x)/g(x) = 0/0 or ±∞/±∞, then lim_(x -> c) f(x)/g(x) can be found by lim_(x -> c) f'(x)/g'(x), provided the latter limit exists.Find Derivatives: We will find the derivatives of the numerator and the denominator: The derivative of the numerator f(x) = sin(7x) - 7x cos(7x) is f'(x) = 7 cos(7x) - 7 cos(7x) + 49x sin(7x). The derivative of the denominator g(x) = 7x - sin(7x) is g'(x) = 7 - 7 cos(7x).Simplify Derivative: Simplify the derivative of the numerator: f'(x) = 7 cos(7x) - 7 cos(7x) + 49x sin(7x) simplifies to f'(x) = 49x sin(7x).Apply Rule Again: Now we apply L'Hôpital's Rule by taking the limit of the derivatives: lim_(x -> 0) f'(x)/g'(x) = lim_(x -> 0) (49x sin(7x)) / (7 - 7 cos(7x))Find New Derivatives: Evaluate the limit of the new expression as x approaches 0: lim_(x -> 0) (49x sin(7x)) = 0 lim_(x -> 0) (7 - 7 cos(7x)) = 0 We again have an indeterminate form of 0/0, so we need to apply L'Hôpital's Rule once more.Evaluate New Expression: Find the derivatives of the new numerator and denominator: The derivative of the new numerator h(x) = 49x sin(7x) is h'(x) = 49 sin(7x) + 343x cos(7x). The derivative of the new denominator k(x) = 7 - 7 cos(7x) is k'(x) = 7 sin(7x).Re-evaluate Derivatives: Apply L'Hôpital's Rule again by taking the limit of the new derivatives: lim_(x -> 0) h'(x)/k'(x) = lim_(x -> 0) (49 sin(7x) + 343x cos(7x)) / (7 sin(7x))Evaluate the limit of the new expression as x approaches 0: lim_(x -> 0) (49 sin(7x) + 343x cos(7x)) = 49 * 0 + 343 * 0 * 1 = 0 lim_(x -> 0) (7 sin(7x)) = 7 * 0 = 0 We still have an indeterminate form of 0/0, which means we made a mistake in our differentiation or simplification. We need to re-evaluate the derivatives.

Use L'Hôpital's Rule (possibly more than once) to evaluate the following limit. (Use symbolic notation and fractions where needed.) $\newline$ $\lim_{x \to 0}\frac{\sin(7x)-7x \cos(7x)}{7x-\sin(7x)}=$

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Use L'Hôpital's Rule (possibly more than once) to evaluate the following limit. (Use symbolic notation and fractions where needed.) $\newline$ $\lim_{x \to 0}\frac{\sin(7x)-7x \cos(7x)}{7x-\sin(7x)}=$