Limits and Derivatives Review

Evaluating Limits

• Direct Evaluation: First approach for evaluating limits. Check for restrictions (division by zero, infinities, etc.).

  • If the function is just a number, the limit is that number.

    • Example: $\lim_{x \to -1} -1 = -1$ because it's just a horizontal line at y = -1.

  • If there are no restrictions, plug in the value directly.

    • Example: $\lim_{x \to -4} -x - 4 = -(-4) - 4 = 4 - 4 = 0$

    • Example: $\lim_{x \to -2} -2x + 3 = -2(-2) + 3 = 4 + 3 = 7$

  • Radicals: Even numbered radicals (square root, fourth root, etc.) require the expression inside to be greater than or equal to zero to avoid imaginary numbers.

    • Example: $\lim<em>{x \to 5} -\sqrt{2x + 1}$. Here $2x + 1 \geq 0$, so $x \geq -\frac{1}{2}$. Since 5 is in the safe territory, this is $\lim</em>{x \to 5} -\sqrt{2(5) + 1} = -\sqrt{11}$.

  • Odd numbered radicals have no restrictions; negative numbers are allowed inside.

    • $\lim_{x \to -2} \sqrt[3]{-2x - 3} = \sqrt[3]{-2(-2) - 3} = \sqrt[3]{4-3} = \sqrt[3]{1} = 1$

  • Using the Unit Circle:

    • Example: $\lim_{x \to \frac{\pi}{6}} \cos(x) = \frac{\sqrt{3}}{2}$ (Cosine at $\frac{\pi}{6}$ is $\frac{\sqrt{3}}{2}$).

Jumps

• For jump discontinuities, the actual limit does not exist. However, one-sided limits can be found.

  • One-Sided Limits: Notation includes a plus (+) for the limit from the positive side (right) and a minus (-) for the limit from the negative side (left).

    • Example (from a graph): $\lim<em>{x \to 3^+} f(x) = 0$ and $\lim</em>{x \to 3^-} f(x) = -6$. Since these are not equal, the actual limit (two-sided) DNE (Does Not Exist).

    • Example (from a graph): $\lim_{x \to 3^-} f(x) = 3$

• Piecewise Functions:

  • If there is no plus or minus sign, a limit from both sides must exist and be equal for the overall limit to exist.

  • Example: Given a piecewise function, if $\lim<em>{x \to -2^-} f(x) = -1$ and $\lim</em>{x \to -2^+} f(x) = -5$, then $\lim_{x \to -2} f(x)$ DNE because the one-sided limits are not equal.

  • For one-sided limits with piecewise functions, determine which piece of the function applies based on the direction of approach.

    • Example: Find $\lim<em>{x \to -1^-} f(x)$ given $f(x) = \begin{cases} x + 2, &amp; x \leq -1 \ -\frac{1}{2}x - 4, &amp; x &gt; -1 \end{cases}$ Since we are approaching from the negative side, we use the top piece, $x + 2$. Plugging in -1 gives $\lim</em>{x \to -1^-} f(x) = -1 + 2 = 1$ .

  • To evaluate, plug the x value into the appropriate equation. Graphing the piecewise function can also help visualize the one-sided limits.

    • Given an X approaching $4$ from the negative side, for a piecewise function with the top piece being $2x - 5$, the limit would be $2(4) - 5 = 3$.

  • Graphing calculators are useful to graph functions, expecially to graph piecewise functions.

Removable Discontinuities:

• At a removable discontinuity, the limit exists if both sides approach the same value, even if the function value at that point is different.

• Example: Even if $f(4) = -1$, if $\lim<em>{x \to 4^-} f(x) = 4$ and $\lim</em>{x \to 4^+} f(x) = 4$, then the full limit $\lim_{x \to 4} f(x) = 4$.

Asymptotes

• If a function has an asymptote at a particular x-value, and the limits from each side go to positive or negative infinity (even if different infinities), the limit at that point DNE (Does Not Exist).

• Example: If one side approaches positive infinity and the other approaches negative infinity, at $x = -4$, the limit DNE (Does Not Exist), even though it may be 1/4.

• If both sides go to the same infinity, the limit is that infinity, like positive infinity.

  • Example: $\lim<em>{x \to -1^-} f(x) = \infty$ and $\lim</em>{x \to -1^+} f(x) = \infty$, therefore, $\lim_{x \to -1} f(x) = \infty$.

• When evaluating one-sided limits at asymptotes, focus on the direction specified (positive or negative side).

  • Example: Even if both sides approach infinity, if the problem asks for the limit as x approaches from the negative side, the answer is that infinity.

Limits Approaching Infinity:

• When x approaches infinity (positive or negative), consider the behavior of y.

• If there is no plus or minus specified, assume it's the positive side.

• Example: As x goes to positive infinity, y approaches 0.

• Example: As x goes to negative infinity, y goes to negative infinity.

• When evaluating algebraically (without a graph):

• Consider what happens to the function as x gets very large.

• Constants become insignificant compared to large x values.

  • Example: Evaluating $\lim_{x \to \infty} \frac{x}{x + 3}$, as x gets very large, the +3 becomes insignificant, and the limit approaches 1.

    • When $x = 1$, $f(x) = \frac{1}{4}$. When $x = 10$, $f(x) = \frac{10}{13}$. When $x = 100$, $f(x) = \frac{100}{103}$. When $x = 1,000,000$, the expression goes to 1.

  • Example: $\lim_{x \to \infty} \frac{x^3}{x^2 - 2}$ approaches infinity because the -2 becomes insignificant. As x increases significantly, the function approaches x, so the limit is infinity.

• If there are no balancing factors, it can probably be assumed it approaches infinity.

  • Example: $\lim_{x \to \infty} x - 1$ also approaches infinity
    *

• If the denominator grows faster than the numerator, the limit approaches zero.

  • Example: $\lim_{x \to -\infty} \frac{-x + 2}{x^2 + x + 1}$ approaches zero because the $x^2$ term in the denominator dominates as x goes to negative infinity.

Instantaneous Speed and Velocity

Average Velocity Formula

• Historically, calculus was developed for physics problems, especially mechanics.

• Velocity is the change in an object over an interval.

  • Average velocity is displacement (change in position) over time.

    • If one travels 100 km from town A to town B in 2 hours, the average velocity is 50 km/h.
      $Average\ Velocity=\frac{\Delta\ position}{time} = \frac{position<em>B - position</em>A}{time<em>B - time</em>A}$

• Speed is the magnitude of velocity, whereas velocity includes direction.

  • Speed is a scalar (magnitude only), while velocity is a vector (magnitude and direction).
    $speed = |velocity|$

  • Common units for speed include meters per second, miles per hour, feet per second, etc.

• Instantaneous speed gets us moving into what a derivative is.

  • Instantaneous speed is speed at an instant (or, speed over an infinitesimally short period of time).

  • To find instantaneous speed at a point, one can use the limit to analyze it.

  • The limit used can shrink the period of time and get it smaller and smaller to figure out what what the change in time is at a certain momment.

  • As a function gets closer and closer to 0, it will look like a line.

• Average speed is what you know, as speed is displacement over time.

• In order to solve a problem, you need to look at smaller and smaller time units.

• As the time units get smaller, the average speed approaches the actual speed, however, the average value is not the actual.

• You can look at these units with the limit.

$\text{average velocity} = \frac{\text{change in distance}}{\text{change in time}} = \frac{\Delta s}{\Delta t}$

Instantaneous Speed Formula

• 
  Formula: $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$, where h is a very small distance.

• This can be used to find displacement fom point A to point B, when A and B are very close together.

• Instantaneous rate of change is essentially the slope of the tangent line at a specific point.

• To calculate instantaneous velocity or the slope of a tangent line on a curve:

  • 
    Find average velocity

  • 
    Make slices smaller and smaller.

Finding Average Velocity

• Average velocity is the change in y over the change in x.

• Instantaneous velocity is the idea, but the slices get smaller and smaller.

• This can be defined as the following formulas; h is really small, but it's used as the same thing.

• Average Velocity Formula using Limit:
  $\lim_{h \to 0} \frac{s(t + h) - s(t)}{h}$

• Algebraic example with $f(x) = 3x^2$: using $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Plug in (x + h) into the formula.

• Simplify, cancelling out terms, and finally, end with $\lim_{h \to 0} -6x + 3h$.

• Then plug in 0, and the final instantaneous rate of change (derivative), which will be equal to -6x.

• Algebraic example with $f(x) = x^2 + x + 1$ and asked to find the instantaneous rate of change fo the function at x = -2.

• Plug in (-2+h) into the equation. (Can also be rewritten as h - 2).

• Simplify, which will come out to $h^2 -3h + 3$.

• Take new equation, and put into limit formula, and simplify.

• Final will be f'(x) = -3.

• Therefore, the derivate at x = -2 is -3.

Definition of Derivative:

• The is a notation to calculate any point for the instantaneous rate of change.

• There are 3 notations that can be used to describe the process, but are all basically the same notation.

  • Newton: f'(x)

  • Leibniz: $\frac{dy}{dx}$

  • Modern: y'

• Basic Formula

```

$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• This definition is the basic derivative definition, and will change how we handle these problems from now on.

• If the limited, the derivate will exist.

Derivative Examples

• Take the function, and plug the new x term into the limit.

• Example with algebra f(x)= 2x+2.

First plug in x+h which you will get 2x = 2(x=h) + 2 = 2x + 2h + 2.

Next, write the formula with the limit as h approaches 0:

$\lim_{h \to 0} \frac{2x+2h+2 - 2x+2}{h}$

Last = 2 to evaluate.
Example problem, f(x) = 4x squared =4 and solve.
First, apply x+h. f(x+h) = 4(x+h)squared + 4 = 4(x squared + 2xh + hquared) + 4
Then
Follow with putting it into that derivative formula (limit and stuff):

$\lim_{h \to 0} \frac{4x squared( + 8xh+4h squared + 4 - 4x squared - 4}{h} = 8x"+4h + 1/2= 8x$