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Problem 14 Medium Difficulty

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.

$ f(x) = 3x^4 - 5x + \sqrt[3]{x^2 + 4}, \hspace{5mm} a = 2 $


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Video Transcript

This is from number fourteen. His tour Calculus, a sedition section two point find. Use the definition of continuity and the properties of Clement's to show that the function is continuous at the Cuban number, eh? If X is equal to three x to the fourth power minus five x plus the cube root of the quantity X x squared plus four for a close to two on. So the general definition for continuity is that element as X approaches A for the function. Yeah, is equal to F A. And so the way that will show this is, too. Calculate the limit and then see if it equals vervain. So we begin with Element A Z X approaches aim, and this case is too, of this entire quantity. Three acts of the fourth minus five x plus the cube root of the quantity X plus. For since we have three terms that are either being added or subjected, willies are some indifference. Longs to separate this and two three different limits. Luminous explosions to two of three x to the fourth power, minus the limited expertise to Dr Nix Place. The limit is six purchase, too, on the Q Bert, the cube root of ex squiggles for which I'm going to write as explosive squared plus four to the wanted power for the first two terms will be able to use our concept multipolar parliament, which means that we can pull out any constant, simple find limit further. They pull out a three from the first function and a fire from the second function here for into third limit, we use the power that's st Stand. If a function is being raised to the power instead of element Reid, you may you may separate. You may raise the limit itself to that power and since we have two term that inside were we'LL also add in used the some alarm to separate this inner a function into to limits and then everything is raised to the one third power. Finally, we really with this limit three times X approaching to race with fourth power and it's five times the limit is expertise to Neck of X is equal to close. The limit is exposed to X squared is two squared plus, the limited exposure is two of four is equal to four. This is always to the one there power appear three ten to the fourth. Power is going to be forty eight Minus. Frightened, too, is ten plus two. Squared applies for is eight, and the Cure eatery is crying, too. So fourteen minus tennis, thirty eight plus two is forty, and we see that free or tap directly plankton. Two friends. We will get the same result. F of two is exactly forty eight minus ten, plus the queue brutally, which is too. And that's also forty. So we have prevented dysfunction is continuous city, um at a equals two for this function?