Questions: 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 + ∛(x^2+4), a=2

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 + ∛(x^2+4), a=2
Transcript text: 14-15 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $a$. 14. $f(x)=3 x^{4}-5 x+\sqrt[3]{x^{2}+4}, \quad a=2$
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Solution

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Solution Steps

To show that a function is continuous at a given point a a , we need to verify that the limit of the function as x x approaches a a is equal to the function's value at a a . This involves checking that the limit from the left and the right both equal f(a) f(a) .

  1. Calculate f(a) f(a) .
  2. Compute the limit of f(x) f(x) as x x approaches a a .
  3. Verify that the limit equals f(a) f(a) .
Step 1: Calculate f(a) f(a)

To find the value of the function at a=2 a = 2 , we compute: f(2)=3(2)45(2)+(2)2+43 f(2) = 3(2)^4 - 5(2) + \sqrt[3]{(2)^2 + 4} Calculating this gives: f(2)=3(16)10+4+43=4810+83=4810+2=40 f(2) = 3(16) - 10 + \sqrt[3]{4 + 4} = 48 - 10 + \sqrt[3]{8} = 48 - 10 + 2 = 40

Step 2: Compute the Limit

Next, we find the limit of f(x) f(x) as x x approaches 2 2 : limx2f(x)=40 \lim_{x \to 2} f(x) = 40

Step 3: Verify Continuity

To show that f(x) f(x) is continuous at a=2 a = 2 , we check if: limx2f(x)=f(2) \lim_{x \to 2} f(x) = f(2) Since both values are equal: 40=40 40 = 40 This confirms that f(x) f(x) is continuous at x=2 x = 2 .

Final Answer

The function f(x) f(x) is continuous at a=2 a = 2 , and thus we conclude: The function is continuous at a=2. \boxed{\text{The function is continuous at } a = 2.}

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