Calculate the limit for the function $f(x)=5 x$ and interval $[0,2]$. Verify your answer by using geometry. (Give your answer as a whole or exact number.)
Solution
Step 1 ：The limit of a function as x approaches a certain value is the value that the function approaches as x gets closer and closer to that value. In this case, we are asked to find the limit of the function \(f(x)=5x\) as x approaches any value in the interval \([0,2]\).
Step 2 ：Since \(f(x)=5x\) is a linear function, its limit at any point in its domain is simply the function value at that point. Therefore, the limit of \(f(x)=5x\) as x approaches any value in the interval \([0,2]\) is \(5x\) for that value of x.
Step 3 ：We can verify this by plotting the function and observing its behavior in the interval \([0,2]\). The plot shows that the function \(f(x)=5x\) is a straight line passing through the origin. As x increases from 0 to 2, the function value increases linearly from 0 to 10.
Step 4 ：This confirms our earlier thought that the limit of \(f(x)=5x\) as x approaches any value in the interval \([0,2]\) is \(5x\) for that value of x.
Step 5 ：\(\boxed{\text{Final Answer: The limit of the function } f(x)=5x \text{ in the interval } [0,2] \text{ is } 5x \text{ for } x \in [0,2]}\)
