Evaluate the Limit limit as x approaches 3 of 2x+|x-3|

Problem

(lim_x→3)(2*x+|x−3|)

Solution

1. Identify the type of limit. Since the function ƒ(x)=2*x+|x−3| is a combination of a linear function and an absolute value function, it is continuous everywhere on the real line.

2. Apply the direct substitution property for continuous functions. To find the limit as x approaches 3 substitute x=3 into the expression.

3. Substitute the value 3 into the expression.

2*(3)+|3−3|

4. Simplify the arithmetic inside and outside the absolute value.

6+|0|

5. Calculate the final result.

6+0=6

Final Answer

(lim_x→3)(2*x+|x−3|)=6

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