Question

Find the indicated limit, if it exists.

limit of f of x as x approaches 0 where f of x equals 10 x plus 2 when x is less than 0 and the absolute value of the quantity 2 minus x when x is greater than or equal to 0

Answer 1

Answer

The limit exists and equals 2.

Explanation

Since f(x)=10x+2 for x$\isin ]-\infin; 0[$ and it is continuous on ]$-\infin$; 0[ as a polynomial, then its left limit at the point x=0 exists and equals f(0-0)=f(0)=10*0+2=2.

On the other hand, f(x)=|2-x| for x $\isin [0,+\infin[$ and it is continuous on [0,+$\infin$[ as the absolute value of a polynomial, then the right limit at the point x=0 equals

f(0+0)= f(0)=|2-0|=2.

Hence, f(0-0)=f(0+0)=2, and the limit of f (x) as x approaches 0 exists and equals 2.