Differential equationsUse the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
8 months ago
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. p(v)=2 sqrt(3v^(2)+1), a=1
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Browhywegottadothis Staff answered 8 months ago
f(x) is continuous at x=a if and only if lim_(x -> a)f(x)=f(a) lim_(v -> 1)p(v)= lim_(v -> 1)2 sqrt(3v^(2)+1) Using the constant law for limits, we can write =2* lim_(v -> 1) sqrt(3v^(2)+1) Use theorem 9 to simplify lim_(x -> 2) sqrt[3](x^(2)+4) =2 sqrt(lim_(v -> 1)3v^(2)+1) Using the sum law for limits, we can write =2 sqrt(lim_(v -> 1)3v^(2)+ lim_(v -> 1)1) Recall that: lim_(x -> a)x^(n)=a^(n) =2 sqrt(3*1^(2)+1)=p(1) Hence proved that p(x) is continuous at x=1 Result: Hint: we have to show that lim_(v -> 1)p(v)=p(1)
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