Question

question
evaluate the limit: \\(\lim\limits_{x\to 0} \dfrac{5x - 5x\cos(x)}{7x\sin(x)}\\)
answer attempt 1 out of 2
dne

Explanation:

Step1: Simplify the numerator

Factor out \(5x\) from the numerator: \(5x - 5x\cos(x)=5x(1 - \cos(x))\)
So the expression becomes \(\lim_{x
ightarrow0}\frac{5x(1 - \cos(x))}{7x\sin(x)}\)

Step2: Cancel out common terms

Cancel out the common factor \(x\) from the numerator and the denominator: \(\lim_{x
ightarrow0}\frac{5(1 - \cos(x))}{7\sin(x)}\)

Step3: Use the trigonometric limit formula

Recall the limit formula \(\lim_{x
ightarrow0}\frac{1 - \cos(x)}{\sin(x)} = 0\) (we can also use L'Hopital's Rule or the identity \(1-\cos(x)=2\sin^{2}(\frac{x}{2})\) to verify).
Substitute this limit into our expression: \(\frac{5}{7}\times\lim_{x
ightarrow0}\frac{1 - \cos(x)}{\sin(x)}\)
Since \(\lim_{x
ightarrow0}\frac{1 - \cos(x)}{\sin(x)} = 0\), we have \(\frac{5}{7}\times0 = 0\)

Answer:

\(0\)