Question

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2

Accepted Answer

Identify the function given: $$ g(	heta) = \frac{5 \cos 	heta}{4	heta - 2\pi} $$ and the point $$ a = \frac{\pi}{2} . $$Check for continuity at $$ a = \frac{\pi}{2}  by $$substituting $$ 	heta = \frac{\pi}{2} $$ into the denominator $$ 4	heta - 2\pi . If $$the denominator equals zero, the function is not defined at this point. Since the denominator becomes zero at $$ 	heta = \frac{\pi}{2} $$, the function is not defined at this point, indicating a potential discontinuity. Use a graphing tool to plot $$ g(	heta) $$ and observe the behavior of the function as $$ 	heta $$ approaches $$ \frac{\pi}{2} $$ from both the left and the right. Determine if the function can be extended to be continuous from either the left or the right by observing the limit of $$ g(	heta)  as$$ $$ 	heta $$ approaches $$ \frac{\pi}{2} . If $$the left-hand limit and right-hand limit are equal, suggest this common value as the extended function's value at $$ a = \frac{\pi}{2} .$$

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Concetti chiave

Continuous Functions

Limits

Right-Hand and Left-Hand Limits