MATH 131, section 1 – Calculus for Physical and Life Sciences First Limit Examples September 24, 2007

Example 1. Investigate the behavior of f(x) = cos(x) for x near a = \frac{π}{3} ≐ 1.0472 radians. Can we “spot the trend?”

\begin{matrix}x & .9 & 1.0 & 1.04 & 1.047 & 1.0471 & 1.048 & 1.05 & 1.1 \\ cos(x) & .62161 & .54030 & .50622 & .50017 & .50008 & .49930 & .49757 & .45360 \end{matrix}

(Note: all values are rounded to 5 decimal places.)

Guess: lim_{x→ \frac{π}{3}} cos(x) = .

Example 2. Investigate the behavior of h(x) = \frac{x^{2}−4x+3}{x−1} for x near a = 1. What is the trend here?

\begin{matrix}x & 0.0 & 0.5 & 0.9 & 0.99 & 1.01 & 1.1 & 1.5 & 2.0 \\ \frac{x^{2}−4x+3}{x−1} & −3.0 & −2.5 & −2.1 & −2.01 & −1.99 & −1.9 & −1.5 & −1 \end{matrix}

Guess: lim_{x→1} \frac{x^{2} − 4x + 3}{x − 1} = .

Example 3. Investigate the behavior of g(x) = x sin (\frac{1}{x}) for x near a = 0. What is the trend here?

\begin{matrix}x & −1.0 & −.1 & −.01 & −.001 & .001 & .01 & .1 & 1.0 \\ x sin (\frac{1}{x}) & .84147 & −.05440 & −.00506 & .00083 & .00083 & −.00506 & −.05440 & .84147 \end{matrix}

Guess: lim_{x→0} x sin (\frac{1}{x}) = .

Here the graph explains the apparent frequent oscillations from positive to negative values and back (turn over for graph).

The graph y = x sin (\frac{1}{x}) for x near 0.

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