Limits of Functions: Tables, Graphs, and Intuitive Definitions

The Limit of a Function

Limits (tables & graphs)

• Consider the function $f(x) = \frac{x-1}{x^2-1}$. The central question is: As the values of x get closer and closer to 1, what do the values of $f(x)$ approach?
• In notation, this is expressed as: $\lim_{x \to 1} \frac{x-1}{x^2-1} = ?$
• To answer this, we can create a table of values for x approaching 1.
• From the table, it's observed that as x approaches 1 from values less than 1 (the left side of 1), $f(x)$ seems to approach 0.5.
• This is written as: $\lim_{x \to 1^-} \frac{x-1}{x^2-1} = 0.5$
  • This is verbalized as: "The left-hand limit of f(x) as x approaches 1 (from the left) is equal to 0.5."
• Similarly, as x approaches 1 from values greater than 1 (the right side of 1, but not equal to 1), the outputs $f(x)$ also appear to approach 0.5.
• This is written as: $\lim_{x \to 1^+} \frac{x-1}{x^2-1} = 0.5$
  • This is verbalized as: "The right-hand limit of f(x) as x approaches 1 (from the right) is equal to 0.5."
• Since the right and left-hand limits are the same, the two-sided limit appears to exist. Thus:
  • $\lim_{x \to 1} \frac{x-1}{x^2-1} = 0.5$
  • or $f(x) \to 0.5$ as $x \to 1$
  • Which reads: "f(x) approaches 0.5 as x approaches 1"

Graphical Understanding

• Graphing the function helps visualize the behavior of $f(x)$ near x = 1.
• Important: In this example, f was undefined at x=1 since $f(1) = \frac{0}{0}$, and that’s why there is a hole in the graph.
• The limit only examines the behavior of f near 1, not at x=1 itself. Values like x=0.999999999999 matter, while x=1 does not.
• Warning: Estimating limits using tables and graphs can be misleading due to the limitations of calculators, graphing tools, and human judgment.
• The next section will cover algebraic methods for precisely calculating limits.

Intuitive Definition of a Limit

• Suppose $f(x)$ is defined when x is near the number a. Then we write $\lim_{x \to a} f(x) = L$
  • and say “the limit of $f(x)$, as x approaches a, equals L” if we can make the values of $f(x)$ arbitrarily close to L (as close as we like) by restricting x to be sufficiently close to a (on either side of a, but $x \neq a$).

Examples

Example 1: $\lim_{x \to 0} \frac{\sin(x)}{x}$

• Based on a table and graph, we estimate $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

Example 2: $\lim_{x \to 0} \sin(\frac{\pi}{x})$

• Initial observation might suggest the limit is 0, but this is incorrect.
• As x approaches 0, $\frac{\pi}{x}$ grows larger, causing $\sin(\frac{\pi}{x})$ to oscillate between -1 and 1.
• Since the function never settles on a single value L, $\lim_{x \to 0} \sin(\frac{\pi}{x})$ does not exist (DNE).

Example 3: Heaviside Function

• Evaluate $\lim<em>{t \to 0^-} H(t)$, $\lim</em>{t \to 0^+} H(t)$, and $\lim_{t \to 0} H(t)$ if

  $H(t) = \begin{cases} 0 &amp; \text{if } t &lt; 0 \ 1 &amp; \text{if } t \geq 0 \end{cases}$

• Solution:

  • If t approaches 0 from the left, then t < 0, so $\lim_{t \to 0^-} H(t) = 0$.
  • If t approaches 0 from the right, then t > 0, so $\lim_{t \to 0^+} H(t) = 1$.
  • Since the left and right-hand limits are different, $\lim_{t \to 0} H(t)$ does not exist (DNE).

• Important Fact: If the left-hand limit ≠ right-hand limit, the two-sided limit DNE.

Example 4: Graphical Evaluation

• Use a given graph to evaluate:
  • $g(2)$, $g(5)$, $\lim<em>{x \to 2^-} g(x)$, $\lim</em>{x \to 2^+} g(x)$, $\lim<em>{x \to 2} g(x)$, $\lim</em>{x \to 5^-} g(x)$, $\lim<em>{x \to 5^+} g(x)$, and $\lim</em>{x \to 5} g(x)$
• Answers:
  • $f(2)$ is undefined (open circles).
  • $f(5) \approx 1.25$
  • $\lim_{x \to 2^-} g(x) = 3$
  • $\lim_{x \to 2^+} g(x) = 1$
  • $\lim_{x \to 2} g(x) = DNE$ (since Right-hand limit ≠ Left-hand limit)
  • $\lim_{x \to 5^-} g(x) = 2$
  • $\lim_{x \to 5^+} g(x) = 2$
  • $\lim<em>{x \to 5} g(x) = 2$ (since Right-hand limit = Left-hand limit, even though $\lim</em>{x \to 5} g(x) \neq g(5) \approx 1.25$)

Infinite Limits

Example 1: $f(x) = \frac{1}{x^2}$

• Question: As x gets closer to 0, what do the values of $f(x)$ approach?
• In other words, find $\lim_{x \to 0} \frac{1}{x^2}$, if it exists.
• If $x = \frac{1}{1000} = 0.001$, then $f(0.001) = f(\frac{1}{1000}) = \frac{1}{(\frac{1}{1000})^2} = \frac{1}{(\frac{1}{1,000,000})} = 1,000,000$
• So, $f(small \ number) = large \ number!$
• As x approaches 0, the values of $f(x)$ grow arbitrarily large, indicating no limit or bound.
• $\lim_{x \to 0} \frac{1}{x^2} = \infty$ (DNE)
• This is verbalized as “the limit of $f(x)$, as x approaches 0, is infinity,” which means the function grows larger and larger and is unbounded.
• Note: For a two-sided limit to exist, it must approach ONE finite number.

Example 2: $f(x) = \frac{1}{x}$

• Question: As x approaches 0 from the left, what do the values of $f(x)$ approach?
• Find $\lim_{x \to 0^-} \frac{1}{x}$, if it exists.
• If x is close to 0 but less than 0, the denominator is a small negative number, making the quotient a large negative number.
• For example, if $x = -0.001$, then $\frac{1}{-0.001} = -1000$
• Thus, $\lim_{x \to 0^-} \frac{1}{x} = -\infty$ (DNE).
• As x approaches 0 from the left, $f(x)$ grows arbitrarily large negative.

Vertical Asymptotes

• Definition: The vertical line x = a is a vertical asymptote of the function $y = f(x)$ if the function has an infinite limit at x = a (left, right, or two-sided).
• Example: The vertical line x = 0 is a vertical asymptote of $f(x) = \frac{1}{x}$ since $\lim_{x \to 0^-} \frac{1}{x} = -\infty$.

More Examples (Infinite Limits)

Example 1: Find $\lim<em>{x \to 3^+} \frac{2x}{x-3}$ and $\lim</em>{x \to 3^-} \frac{2x}{x-3}$

• For $\lim_{x \to 3^+} \frac{2x}{x-3}$: If x is close to 3 but larger than 3, the denominator is a small positive number, and the numerator is close to 6. So the quotient is a large positive number.
  • For example, if $x = 3.001$, then $\frac{2(3.001)}{3.001 - 3} = \frac{6.002}{0.001} = 6002$
  • Thus, $\lim_{x \to 3^+} \frac{2x}{x-3} = \infty$ (DNE).
  • In summary: $\frac{2x}{x-3} \to \frac{6}{0^+} \to \infty$
• For $\lim_{x \to 3^-} \frac{2x}{x-3}$: If x is close to 3 but smaller than 3, the denominator is a small negative number, and the numerator is close to 6. So the quotient is a large negative number.
  • For example, if $x = 2.999$, then $\frac{2(2.999)}{2.999 - 3} = \frac{5.998}{-0.001} = -5998$
  • Thus, $\lim_{x \to 3^-} \frac{2x}{x-3} = -\infty$ (DNE).

Example 2: Find $\lim_{x \to \frac{\pi}{2}^+} \tan(x)$

• Since $\tan(x) = \frac{\sin(x)}{\cos(x)}$, if x is close to $\frac{\pi}{2}$ but larger than $\frac{\pi}{2}$, the numerator is close to $\sin(\frac{\pi}{2}) = 1$, and the denominator, $\cos(x)$, is a small negative number.
• So, the quotient is a large negative number.
• Thus, $\lim_{x \to \frac{\pi}{2}^+} \tan(x) = -\infty$ (DNE).
• In summary: $\tan(x) = \frac{\sin(x)}{\cos(x)} \to \frac{1}{0^-} \to -\infty$

Example 3: Find $\lim_{x \to 0^+} \ln(x)$

• By creating a table or graphing the function, we see $\lim_{x \to 0^+} \ln(x) = -\infty$

Review Question

• Name 3 general situations in which the limit fails to exist.