Question

Help me pleasseeeee!!! its about algebraic properties of limits!

Answer 1

$\text{Hello there! :)}$

$3. \sqrt{2} \\\\\text{4. 0}\\\\\text{5. dne}$

$\text{To find the overall limit, they must approach the same y-value from each side:}$

$\\\\3. \lim_{x \to \frac{\pi }{4} } 2sinx ={\sqrt{2}$

$\text{Therefore:}\\\\3. \lim_{x \to \frac{\pi }{4} } f(x)= {\sqrt{2} }$

$\text{For this question, evaluate the limit from the RIGHT-HAND side:}\\\\4. \lim_{x \to \pi +} f(x)\\\\ \text{Use the equation tanx to evaluate since it involves values greater than \pi:}$

$\lim_{x \to \pi +} tanx = 0\\\\\text{So:}\\\\ \lim_{x \to \pi +} f(x) = 0$

$5. \\\\\text{Evaluate the overall limit. Make sure both sides approach the same y=value:}\\\\ \lim_{x \to \pi } 2cosx= -2\\\\ \lim_{x \to \pi } tanx = 0\\\\ \text{ Therefore:}\\ \lim_{x \to \pi } f(x) = dne \text{ (Does not exist})$