The expression in the image you've provided is: \[ \lim_{x \to -\infty} \frac{\sin x}{7\cos x} \] To solve this limit, let's proceed step by step. ### Step 1: Simplify the Expression The expression can be simplified by factoring out constants: \[ \lim_{x \to -\infty} \frac{\sin x}{7\cos x} = \frac{1}{7} \lim_{x \to -\infty} \frac{\sin x}{\cos x} \] ### Step 2: Recognize the Trigonometric Identity The ratio \(\frac{\sin x}{\cos x}\) is a standard trigonometric identity, known as the tangent function: \[ \frac{\sin x}{\cos x} = \tan x \] So the limit becomes: \[ \frac{1}{7} \lim_{x \to -\infty} \tan x \] ### Step 3: Analyze the Behavior of \(\tan x\) as \(x \to -\infty\) The tangent function, \(\tan x\), is periodic with a period of \(\pi\) and does not have a limit as \(x \to -\infty\). It oscillates between \(-\infty\) and \(+\infty\) infinitely many times because of its periodic nature. Therefore, the limit: \[ \lim_{x \to -\infty} \tan x \] **does not exist** because \(\tan x\) does not converge to a single value as \(x\) approaches \(-\infty\). ### Conclusion: Since the limit of \(\tan x\) as \(x \to -\infty\) does not exist, the original limit: \[ \lim_{x \to -\infty} \frac{\sin x}{7\cos x} \] **does not exist**. Would you like further details or have any other questions? Here are some related questions you might find interesting: 1. What is the behavior of the \(\tan x\) function as \(x\) approaches specific values? 2. How do periodic functions affect limits at infinity? 3. What happens to trigonometric functions as \(x\) approaches infinity? 4. Can limits of trigonometric functions exist under different circumstances? 5. How does the period of the \(\tan x\) function influence its limit behavior? 6. How do we handle limits of functions that oscillate? 7. What is the significance of the tangent function in trigonometry? 8. How does the presence of constants affect limits in trigonometric functions? **Tip:** When dealing with trigonometric limits, always consider the periodic nature of the function, as it can significantly impact whether a limit exists. replay 7 with x and then translate to vietnamese