Neetesh Kumar | October 15, 2024
Calculus Homework Help
Assignment Question on Continuous Function
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Step-by-step solution:
To ensure that f(x) is continuous at x=c, the function values from both sides of c must be equal at x=c. Specifically, we need:
limx→c−f(x)=limx→c+f(x)=f(c)
Step 1: Evaluate the left-hand limit as x→c−
For x≤c, the function is given by f(x)=x2−8. So, the left-hand limit as x→c− is:
limx→c−f(x)=c2−8
Step 2: Evaluate the right-hand limit as x→c+
For x>c, the function is given by f(x)=2x−9. So, the right-hand limit as x→c+ is:
limx→c+f(x)=2c−9
Step 3: Set the limits equal to each other
To ensure continuity at x=c, we set the left-hand and right-hand limits equal to each other:
c2−8=2c−9
Step 4: Solve for c
Simplify the equation:
c2−8=2c−9
Rearrange the terms:
c2−2c+1=0
This is a quadratic equation. We can factor it:
(c−1)2=0
Thus, c=1.
Therefore, the value of c that makes the function continuous is:
c=1
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