Properties of Limits

So far we have concentrated on limits of single functions and elementary combinations of functions. If we are working with limits of other combinations or compositions of functions, the situation is slightly more difficult, but sometimes these more complicated limits have useful geometric interpretations. 

Example 1: Use the function defined by the graph in Fig. 1 to evaluate

Solution: (a) $\lim\limits_{x \rightarrow 1}\{3+\mathrm{f}(\mathrm{x})\}$ is a straightforward application of part (a) of the Main Limit Theorem:

$\lim\limits_{x \rightarrow 1}\{3+f(x)\}=\lim\limits_{x \rightarrow 1} 3+\lim\limits_{x \rightarrow 1} f(x)=3+2=5$

(b) We first need to examine what happens to the quantity $2+x$, as $x \rightarrow 1$, before we can consider the limit of $\mathrm{f}(2+\mathrm{x})$. When $\mathrm{x}$ is very close to 1, the value of $2+x$ is very close to 3, so the limit of $f(2+x)$ as $x \rightarrow 1$ is equivalent to the limit of $\mathrm{f}(\mathrm{w})$ as $\mathrm{w} \rightarrow 3 \quad(\mathrm{w}=2+\mathrm{x})$, and it is clear from the graph that $\lim\limits_{x \rightarrow 3} \mathrm{f}(\mathrm{w})=1: \quad \mathrm{f}(2+\mathrm{x})=\lim\limits_{x \rightarrow 3} \mathrm{f}(\mathrm{w})=1 \quad$ (w represents $\left.2+\mathrm{x}\right)$.

In most cases it is not necessary to formally substitute a new variable $\mathrm{w}$ for the quantity $2+\mathrm{x}$, but it is still necessary to think about what happens to the quantity $2+x$ as $x \rightarrow 1$.

(c) As $x \rightarrow 0$, the quantity $3-x$ will approach 3 so we want to know what happens to the values of $f$ when the variable is approaching 3: $\lim\limits_{x \rightarrow 0} \mathrm{f}(3-\mathrm{x})=1$.

(d) $\lim\limits_{x \rightarrow 2}\{\mathrm{f}(\mathrm{x}+1)-\mathrm{f}(\mathrm{x})\}=\lim\limits_{x \rightarrow 2} \mathrm{f}(\mathrm{x}+1)-\lim\limits_{x \rightarrow 2} \mathrm{f}(\mathrm{x})$ replace $\mathrm{x}+1$ with $w$ 

$=\lim\limits_{x \rightarrow 3} f(w)-\lim\limits_{x \rightarrow 2} f(x)=1-3=-2$

Practice 3: Use the function defined by the graph in Fig. 2 to evaluate

(a) $\lim\limits_{x \rightarrow 1} \mathrm{f}(2 \mathrm{x})$

(b) $\lim\limits_{x \rightarrow 2} \mathrm{f}(\mathrm{x}-1)$

(c) $\lim\limits_{x \rightarrow 0} 3 \cdot \mathrm{f}(4+\mathrm{x})$

(d) $\lim\limits_{x \rightarrow 2} f(3 x-2)$.

Example 2: Use the function defined by the graph in Fig. 3 to evaluate

(a) $\lim\limits_{h \rightarrow 0} \mathrm{f}(3+\mathrm{h})$ 

(b) $\lim\limits_{h \rightarrow 0} \mathrm{f}(3)$

(c) $\lim\limits_{h \rightarrow 0}\{\mathrm{f}(3+\mathrm{h})-\mathrm{f}(3)\}$

(d) $\lim\limits_{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}$

Solution: Part (d) is a common form of limit, and parts (a) - (c) are the steps we need to evaluate (d).

(a) As $h \rightarrow 0$, the quantity $w=3+h$ will approach 3 so $\lim\limits_{h \rightarrow 0} f(3+h)=\lim\limits_{x \rightarrow 3} f(w)=1$.

(b) $\mathrm{f}(3)$ is the constant 1 and $\mathrm{f}(3)$ does not depend on $\mathrm{h}$ in any way so $\lim\limits_{h \rightarrow 0} \mathrm{f}(3)=1$.

(c) The limit in part (c) is just an algebraic combination of the limits in (a) and (b):

$\lim\limits_{h \rightarrow 0}\{f(3+h)-f(3)\}=\lim\limits_{h \rightarrow 0} f(3+h)-\lim\limits_{h \rightarrow 0} f(3)=1-1=0$

The quantity $f(3+h)-f(3)$ also has a geometric interpretation - it is the change in the $y$-coordinates, the $\Delta y$, between the points $(3, f(3))$ and $(3+h, f(3+h))$. (Fig. 4)

(d) As $\mathrm{h} \rightarrow 0$, the numerator and denominator of $\frac{\mathrm{f}(3+\mathrm{h})-\mathrm{f}(3)}{\mathrm{h}}$ both approach 0 so we cannot immediately determine the value of the limit. But if we recognize that $f(3+h)-f(3)=\Delta y$ for the two points $(3, f(3))$ and $(3+h, f(3+h))$ and that $h=\Delta x$ for the same two points, then we can interpret $\frac{\mathrm{f}(3+\mathrm{h})-\mathrm{f}(3)}{\mathrm{h}}$ as $\frac{\Delta y}{\Delta x}$ which is the slope of the secant line through the two points so

$\lim\limits_{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}=\lim\limits_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim\limits_{\Delta x \rightarrow 0}\{ \text{slope of the secant line\}}$

$=$ slope of the tangent line at $(3, \mathrm{f}(3)) \approx-2$.

This limit, representing the slope of line tangent to the graph of $\mathrm{f}$ at the point $(3, \mathrm{f}(3))$, is a pattern we will see often in the future.