Find \( \lim _{x \rightarrow-5} \cos (-2 \pi x) \). \[ \lim _{x \rightarrow-5} \cos (-2 \pi x)= \] \( \square \) Show your work and explain, in your own words, how you arrived at your answer. \( \square \) Calculate the limit lim \( \cos (-2 \mathrm{x} x) \)

To find the limit \( \lim_{x \rightarrow -5} \cos(-2 \pi x) \), we can start by substituting \(-5\) directly into the expression, since cosine is continuous everywhere. First, we calculate \(-2 \pi (-5)\): \[ -2 \pi (-5) = 10 \pi. \] Now we evaluate the cosine function at that value: \[ \cos(10 \pi). \] Cosine has a periodicity of \(2\pi\), meaning that \( \cos(x) = \cos(x + 2k\pi) \) for any integer \(k\). Thus, we can simplify: \[ \cos(10 \pi) = \cos(10 \pi - 6\pi) = \cos(4 \pi) = \cos(0) = 1. \] Thus, we have: \[ \lim_{x \rightarrow -5} \cos(-2 \pi x) = 1. \] So the final answer is: \[ \lim_{x \rightarrow -5} \cos(-2 \pi x) = 1. \] To summarize, after substituting \(-5\) into the function and using the periodicity of the cosine function, I concluded that the limit evaluates to 1. \(\square\) 1. In trigonometry, the cosine function is essential because it appears in various real-world situations, such as calculating distances and angles. It helps us model cyclical phenomena—think of the beats of a metronome, where the oscillation mimics the properties of the cosine wave! 2. For those diving deeper into calculus, understanding continuity is key! It’s crucial because many limits can be evaluated simply by direct substitution, especially with continuous functions like polynomials and trigonometric functions. So always check for discontinuities first!