# Determine each of the following limits, if they exist. Briefly justify each of your answers. (a)...

## Question:

Determine each of the following limits, if they exist. Briefly justify each of your answers.

(a) {eq}\lim\limits_{t \rightarrow -2} q(t) {/eq}

(b) {eq}\lim\limits_{h \rightarrow 0} \frac{b(4 + h) - b(4)}{h} {/eq} where {eq}b(x) = 2x + 4 {/eq}

## One-Sided Limits:

In a function given by parts, that is, with different analytical expressions on each side of the point, it is necessary to apply the one-sided limits, if these limits are different, we cannot assure the existence of the limit.

(a) According to the graph we can see the limit for the left and the limit for the right at -1 are different, so the limit {eq}\lim\limits_{t \rightarrow -2} q(t) {/eq} does not exist.

(b) Applying the limit for the given function:

{eq}b\left( x \right) = 2x + 4\\ \mathop {\lim }\limits_{h \to 0} \frac{{b(4 + h) - b(4)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {2(4 + h) + 4} \right) - \left( {2 \cdot 4 + 4} \right)}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {8 + 2h + 4} \right) - \left( {8 + 4} \right)}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \frac{{8 + 2h + 4 - 8 - 4}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \frac{{2h}}{h}\\ = \mathop {\lim }\limits_{h \to 0} 2 = 2 {/eq}