The problem as written is continuous at ##x=0##, so just plug it in. Or if both terms in the denominator are supposed to be cube roots, try L'Hospital's rule.Bunny-chan said:
I assume you meant to write ##\sqrt[3]{x+\pi}## instead of ##3 \sqrt{x+\pi}## in the denominator. If you do not want to (or are unable to) use calculus, use instead the algebraic identity ##a^3-b^3 = (a-b)(a^2+a b + b^2)## for appropriate ##a## and ##b##.Bunny-chan said:
The limit of a trigonometric function can be determined by evaluating the function at the given value and observing the behavior of the function as the input approaches that value. If the function approaches a specific value or oscillates between two values, then that is the limit. If the function does not approach any specific value, then the limit does not exist.
A one-sided limit only considers the behavior of the function as the input approaches the given value from one direction, either from the left or the right. A two-sided limit considers the behavior of the function as the input approaches the given value from both directions. In some cases, the one-sided and two-sided limits may be equal, but in others, they may be different.
In most cases, a trigonometric limit cannot be evaluated algebraically. Instead, it requires the use of trigonometric identities and properties to simplify the function and then evaluate the limit using the rules of limits. However, there are some special cases where the limit can be evaluated algebraically, such as when the function is a constant or a simple rational function.
Yes, it is possible for a trigonometric limit to be undefined. This occurs when the function does not approach any specific value as the input approaches the given value. Instead, the function may oscillate between different values or have a discontinuity at that point, making the limit undefined.
Graphs can be used to visualize the behavior of a trigonometric function and help determine its limit. By plotting the function and observing its behavior near the given value, you can make an educated guess about the limit. Additionally, you can use technology to plot the function and zoom in on the given value to get a more accurate estimate of the limit.