Chapter 2 Test Review: Limits and Continuity
Get ready for your Chapter 2 test! This review will cover key concepts related to limits and continuity, with examples to help you succeed. Remember to stay positive and approach each problem with confidence!
Key Concepts and Examples
1. Limits: Numerical Approach
We can explore limits by examining values of a function as $x$ gets closer and closer to a specific value. For example, let's look at the function $f(x) = \frac{3}{4-x}$ as $x$ approaches 5. We can analyze values near 5:
- As $x$ approaches 5 from the left:
- $x = 4.9$, $f(x) = -3.3333333$
- $x = 4.99$, $f(x) = -3.030303$
- $x = 4.999$, $f(x) = -3.003003$
- As $x$ approaches 5 from the right:
- $x = 5.001$, $f(x) = -2.997003$
- $x = 5.01$, $f(x) = -2.970297$
- $x = 5.1$, $f(x) = -2.7272727$
From this, we can see that the slope of the secant line $PQ$ approaches 3 as $x$ approaches 5. We can write the equation of the tangent line at $x=5$ as $y = 3x - 18$.
2. Limits: Graphical Approach
We can also analyze limits graphically. Consider a function $h(x)$ and its graph. We can determine the limit as $x$ approaches a value $a$ from the left (denoted as $\lim_{x \to a^-} h(x)$) and from the right (denoted as $\lim_{x \to a^+} h(x)$). If these two limits are equal, then the limit exists at $x=a$.
- If $\lim_{x \to 0^-} h(x) = 1$ and $h(0) = 1$, but $\lim_{x \to 0^+} h(x) = -1$, then $\lim_{x \to 0} h(x)$ DNE (Does Not Exist).
- If $\lim_{x \to -3^-} h(x) = 4$, but $h(-3)$ DNE, then the limit exists, but the function is not continuous.
3. Evaluating Limits Analytically
Let's look at some limit evaluation techniques. For example, consider finding the limit of $\lim_{x \to 0} cos(x^3 + 3x)$. Since the cosine function is continuous, we can directly substitute $x=0$:
$$cos(0^3 + 3(0)) = cos(0) = 1$$For more complex limits, we might need to use algebraic manipulation. For example, consider the limit $\lim_{x \to 5} \frac{9-(x+4)}{(x-5)(3+\sqrt{x+4})}$. After simplification, we get:
$$\lim_{x \to 5} \frac{-1}{3 + \sqrt{x+4}} = -\frac{1}{6}$$4. One-Sided Limits and Piecewise Functions
For piecewise functions, we need to consider one-sided limits. For instance, given the function:
$$f(x) = \begin{cases} \sqrt{-x} & \text{if } x < 0 \\ 3 - x & \text{if } 0 \le x < 3 \\ (x - 3)^2 & \text{if } x > 3 \end{cases}$$We can evaluate the following:
- $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (3-x) = 3$
- $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \sqrt{-x} = 0$
- $\lim_{x \to 0} f(x)$ DNE because the left and right limits are not equal.
The function is discontinuous at $x=0$ and $x=3$.
5. Absolute Value and Limits
When dealing with absolute values in limits, we need to be careful about the sign. For example:
$$\lim_{x \to 0} \frac{|8x - 7| - |8x + 7|}{x}$$For $x$ close to 0, we have $8x - 7 < 0$ and $8x + 7 > 0$. Therefore, we can rewrite the limit as:
$$\lim_{x \to 0} \frac{-(8x - 7) - (8x + 7)}{x} = \lim_{x \to 0} \frac{-16x}{x} = -16$$6. Limits Involving Infinity
Let's look at limits as $x$ approaches infinity. For example, find $\lim_{x \to \infty} (x - \sqrt{x})$. In this case, we can see that $x$ grows faster than $\sqrt{x}$, so the limit is infinity.
$$\lim_{x \to \infty} (x - \sqrt{x}) = \infty$$Another example: Find $\lim_{x \to -\infty} \frac{\sqrt{1+4x^6}}{2-x^3}$. This limit can be solved by dividing both the numerator and denominator by $x^3$.
7. Asymptotes
Vertical asymptotes occur where the function approaches infinity (or negative infinity). Horizontal asymptotes occur as $x$ approaches infinity (or negative infinity). Identify these by looking at the graph of the function.
Keep practicing, and you'll do great on your test!