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
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. $f(x)=3 x^{4}-5 x+\sqrt[3]{x^{2}+4}, \quad a=2$
Solution
Solution Steps
To show that a function is continuous at a given point a, we need to verify that the limit of the function as x approaches a is equal to the function's value at a. This involves checking that the limit from the left and the right both equal f(a).
Calculate f(a).
Compute the limit of f(x) as x approaches a.
Verify that the limit equals f(a).
Step 1: Calculate f(a)
To find the value of the function at a=2, we compute:
f(2)=3(2)4−5(2)+3(2)2+4
Calculating this gives:
f(2)=3(16)−10+34+4=48−10+38=48−10+2=40
Step 2: Compute the Limit
Next, we find the limit of f(x) as x approaches 2:
x→2limf(x)=40
Step 3: Verify Continuity
To show that f(x) is continuous at a=2, we check if:
x→2limf(x)=f(2)
Since both values are equal:
40=40
This confirms that f(x) is continuous at x=2.
Final Answer
The function f(x) is continuous at a=2, and thus we conclude:
The function is continuous at a=2.