Question

Evaluate the following limit: \[ \lim_{\theta \to -\frac{\pi}{4}} \frac{\cos\theta + \sin\theta}{\theta + \frac{\pi}{4}}. \]

• A. \( 0 \)
• B. \( 1 \)
• C. \( \sqrt{2} \)
• D. \( \frac{1}{\sqrt{2}} \)

Hint:

For limits involving trigonometric functions, use appropriate substitutions and trigonometric identities to simplify the expression before evaluating the limit.

Answer 1

The Correct Option is C

We are tasked with evaluating the following limit: \[ \lim_{\theta \to -\frac{\pi}{4}} \frac{\cos\theta + \sin\theta}{\theta + \frac{\pi}{4}}. \] Step 1: Simplify the denominator. Let \( \theta + \frac{\pi}{4} = h \). As \( \theta \to -\frac{\pi}{4} \), we have \( h \to 0 \), and the expression becomes: \[ \lim_{h \to 0} \frac{\cos\left(\frac{\pi}{4} - h\right) + \sin\left(\frac{\pi}{4} - h\right)}{h}. \] Step 2: Apply trigonometric identities to simplify the numerator. Using the angle subtraction identities for cosine and sine: \[ \cos\left(\frac{\pi}{4} - h\right) = \cos\frac{\pi}{4}\cos h + \sin\frac{\pi}{4}\sin h, \] \[ \sin\left(\frac{\pi}{4} - h\right) = \sin\frac{\pi}{4}\cos h - \cos\frac{\pi}{4}\sin h. \] Adding these two expressions together: \[ \cos\left(\frac{\pi}{4} - h\right) + \sin\left(\frac{\pi}{4} - h\right) = \left(\cos\frac{\pi}{4} + \sin\frac{\pi}{4}\right)\cos h + \left(\sin\frac{\pi}{4} - \cos\frac{\pi}{4}\right)\sin h. \] Since \( \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}} \), the expression simplifies to: \[ \cos\left(\frac{\pi}{4} - h\right) + \sin\left(\frac{\pi}{4} - h\right) = \sqrt{2}\cos h. \] Step 3: Compute the limit. Substituting the simplified numerator back into the limit: \[ \lim_{h \to 0} \frac{\sqrt{2}\cos h}{h}. \] As \( h \to 0 \), \( \cos h \to 1 \), so the limit evaluates to: \[ \sqrt{2}. \] Final Answer: The value of the limit is: \[ \boxed{\sqrt{2}}. \]