Question

Use the given graph off and g to evaluate the following functions and limits, if it exists. If the limit does not exist, write DNE and explain why. 

(a) \(\lim _{x \rightarrow 0^{+}} f(x)=\)

(b) \(\lim _{x \rightarrow-1} f(x)=\)

(c) \(\lim _{x \rightarrow 1} g(x)=\)

(d) \(\lim _{x \rightarrow-2}\left(\frac{f(x)}{g(x)}\right)=\)

(e) \(\lim _{x \rightarrow 1}(f(x) \cdot g(x))=\)

(f) \(g(-1)=\)

(e) \(\lim _{x \rightarrow 3} f(x)=\)

(h) \(f(0)=\)

Explanation for non existent limits: Explanation is required for credit (i.e., a correct DNE response above is worth no points).

Answer 1

From the right side the function value tends to one.

Both right hand limit and left limit exist and they are equal to zero. Note that the f(-1)=1, different from the limiting value. Thus, the function is not continuous.

The left limit (zero) and right limit (1.5) exist but not equal. Thus the limit is not defined.

Since g(x) has valid right and left limits and f tends to zero as x tends to 1, the product has both the one sided limits equal to zero. Thus the limit exist and is equal to zero.

The one sided limits are not finite.