7. Evaluate the following limits: a) \( \lim _{x \rightarrow 0} \frac{1-\cos (x)}{2 x} \) b) \( \lim _{x \rightarrow 0} \frac{1-\cos (x)}{\sin (x)} \) c) \( \lim _{x \rightarrow 0} \frac{\sin (7 x)}{\sin (5 x)} \) d) \( \lim _{x \rightarrow 2} \frac{\sin (x-2)}{x^{2}-4} \).

Let's evaluate each of the limits one by one. ### a) \[ \lim _{x \rightarrow 0} \frac{1-\cos (x)}{2 x} \] Using the identity \(1 - \cos(x) = 2 \sin^2(\frac{x}{2})\), we can rewrite the limit: \[ \frac{1 - \cos(x)}{2x} = \frac{2 \sin^2(\frac{x}{2})}{2x} = \frac{\sin^2(\frac{x}{2})}{x} \] As \(x\) approaches 0, \(\frac{x}{2} \to 0\) as well, so we can use the small-angle approximation \(\sin(u) \approx u\): \[ = \lim_{x \to 0} \frac{\sin^2(\frac{x}{2})}{x} = \lim_{u \to 0} \frac{u^2}{2u} = \lim_{u \to 0} \frac{u}{2} = 0 \] Thus: \[ \text{Result: } 0 \] --- ### b) \[ \lim _{x \rightarrow 0} \frac{1-\cos (x)}{\sin (x)} \] Again, using the identity \(1 - \cos(x) = 2 \sin^2(\frac{x}{2})\): \[ \frac{1 - \cos(x)}{\sin(x)} = \frac{2 \sin^2(\frac{x}{2})}{\sin(x)} \] Using \(\sin(x) = 2 \sin(\frac{x}{2}) \cos(\frac{x}{2})\): \[ \frac{2 \sin^2(\frac{x}{2})}{2 \sin(\frac{x}{2}) \cos(\frac{x}{2})} = \frac{\sin(\frac{x}{2})}{\cos(\frac{x}{2})} \] As \(x\) approaches 0, \(\frac{x}{2} \to 0\) which leads to: \[ \frac{\frac{x}{2}}{1} = 0 \] Thus: \[ \text{Result: } 0 \] --- ### c) \[ \lim _{x \rightarrow 0} \frac{\sin (7 x)}{\sin (5 x)} \] Using L'Hôpital's Rule since both the numerator and denominator approach 0: \[ = \lim_{x \to 0} \frac{7 \cos(7x)}{5 \cos(5x)} = \frac{7\cos(0)}{5\cos(0)} = \frac{7}{5} \] Thus: \[ \text{Result: } \frac{7}{5} \] --- ### d) \[ \lim _{x \rightarrow 2} \frac{\sin (x-2)}{x^{2}-4} \] We can rewrite the denominator as \(x^2 - 4 = (x - 2)(x + 2)\): \[ \lim _{x \to 2} \frac{\sin(x-2)}{(x - 2)(x + 2)} \] As \(x \to 2\): \[ = \lim_{x \to 2} \frac{\sin(x-2)}{x-2} \cdot \frac{1}{x + 2} \] From the small-angle limit, \(\frac{\sin(u)}{u} \to 1\) as \(u \to 0\): \[ = 1 \cdot \frac{1}{4} = \frac{1}{4} \] Thus: \[ \text{Result: } \frac{1}{4} \] --- In summary, the limits are: a) 0, b) 0, c) \(\frac{7}{5}\), d) \(\frac{1}{4}\).