2.2 Notes on Limits and the Limit of a Function

2 Limits and Derivatives

• This section introduces limits as a fundamental concept for evaluating tangents, velocities, and more generally how a function behaves as the input approaches a point.
• The material distinguishes between numerical/graphical intuition and formal definitions, and builds toward one-sided and infinite limits, plus vertical asymptotes.

2.2 The Limit of a Function

• Intuitive idea (Intuitive Definition of a Limit):
  • If f(x) is defined near a (on some open interval around a, possibly excluding a itself), then the limit of f(x) as x approaches a is L, written as
  • $\lim_{x\to a} f(x) = L$
  • This means we can make f(x) arbitrarily close to L by taking x sufficiently close to a (from either side) but with x ≠ a.
  • Key point: the value of f at x = a is irrelevant for the limit; f(x) need not even be defined at a.
• Alternative notation: the phrase "f(x) approaches L as x approaches a" is another way to express the same idea, written as
  • $f(x) \to L \quad\text{as}\quad x \to a.$
• One can restrict attention to x approaching a from either side, leading to one-sided limits (Definition 2):
  • Left-hand limit: $\lim_{x \to a^-} f(x) = L$ (consider only x < a)
  • Right-hand limit: $\lim_{x \to a^+} f(x) = L$ (consider only x > a)
• The overall limit exists only if both one-sided limits exist and are equal:
  • $\lim<em>{x \to a} f(x) = L\quad\text{iff}\quad \lim</em>{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L.$
• Practical implication: The function value at a (or even the function’s definition at a) is not required for the limit to exist.

Finding Limits Numerically and Graphically

• Example (illustrative function near a = 1):

  • Consider a function f(x) with values near x = 1 that appear to approach 0.5 from both sides.
  • Conclusion:
  • $\lim_{x\to 1} f(x) = 0.5.$
  • Note: The exact algebraic form of f is not necessary to determine the limit from the behavior of nearby values.

• Notation and takeaway:

  • Numerical tables or graphs can reveal the limit by showing f(x) getting arbitrarily close to L as x → a.
  • The limit is about behavior near a, not the exact value at a.

• Example 2: Limit of sin x over x as x approaches 0

  • The function is defined as $f(x) = \frac{\sin x}{x}.$ The function is not defined at x = 0, but the limit as x → 0 exists and equals 1.
  • Numerical illustration (values for x close to 0):
  • $\frac{\sin(1.0)}{1.0} = 0.84147098$
  • $\frac{\sin(0.5)}{0.5} = 0.95885108$
  • $\frac{\sin(0.4)}{0.4} = 0.97354586$
  • $\frac{\sin(0.3)}{0.3} = 0.98506736$
  • $\frac{\sin(0.2)}{0.2} = 0.99334665$
  • $\frac{\sin(0.1)}{0.1} = 0.99833417$
  • $\frac{\sin(0.05)}{0.05} = 0.99958339$
  • $\frac{\sin(0.01)}{0.01} = 0.99998333$
  • $\frac{\sin(0.005)}{0.005} = 0.99999583$
  • $\frac{\sin(0.001)}{0.001} = 0.99999983$
  • Therefore: $\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1.$

• One of the key ideas: limits describe the behavior of f near a, not necessarily at a or at points where the function is undefined.

One-Sided Limits

• Heaviside function as an illustrative example:

  • Heaviside function H is defined by
  • $H(t) = \begin{cases} 0, &amp; t &lt; 0 \ 1, &amp; t \ge 0 \end{cases}.$
  • As t → 0 from the left: $\lim_{t \to 0^-} H(t) = 0.$
  • As t → 0 from the right: $\lim_{t \to 0^+} H(t) = 1.$

• Notation for one-sided limits:

  • Left-hand limit: $\lim_{x \to a^-} f(x) = L.$
  • Right-hand limit: $\lim_{x \to a^+} f(x) = L.$

• Key relationship: a two-sided limit exists iff both one-sided limits exist and are equal.

• Example notation (as in the text):

  • The arrows "-" and "+" indicate left/right direction in approaching a.

• Example 4 (limits from a graph):

  • Given a graph of g, values approach different numbers from left and right at a = 2:
  • Left-hand limit: $\lim_{x \to 2^-} g(x) = 3.$
  • Right-hand limit: $\lim_{x \to 2^+} g(x) = 1.$
  • Since the left and right limits differ, the two-sided limit does not exist:
  • $\lim_{x \to 2} g(x) \text{ does not exist.}$

• Scaling the function preserves the limit if the same one-sided behavior is scaled consistently:

  • For the function 5g(x):
  • $\lim_{x \to 2^-} 5g(x) = 2,$
  • $\lim_{x \to 2^+} 5g(x) = 2.$
  • Since both one-sided limits are equal,
  • $\lim_{x \to 2} 5g(x) = 2.$
  • Note: Even though the limit exists for 5g(x), the original value g(2) may be different (g(5) ≠ 2 in the example).

How Can a Limit Fail to Exist?

• A limit can fail to exist in multiple ways:

  • If the left-hand and right-hand limits are not equal (as in Example 4).
  • If the function oscillates infinitely often as x approaches a, so it does not settle to any single value (Example 5).

• Example 5: Investigate $\lim_{x \to 0} \sin\left(\frac{p}{x}\right).$

  • For a fixed nonzero p, as x → 0, the argument p/x oscillates without bound, so sin(p/x) oscillates between -1 and 1.
  • The function does not approach a single value; hence the limit does not exist.
  • Intuition: There are sequences xk → 0 for which sin(p/xk) takes values 1, -1, 0, etc., infinitely often as x approaches 0.

Infinite Limits; Vertical Asymptotes

• Intuitive definition of an infinite limit:
  • If f is defined on both sides of a (except possibly at a), then
  • $\lim_{x \to a} f(x) = \infty$ means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a (but not equal to a).
  • Another notation: $f(x) \xrightarrow{x \to a} \infty.$
• The same idea with negative infinity:
  • $\lim_{x \to a} f(x) = -\infty$ means f(x) can be made arbitrarily large in magnitude in the negative direction as x → a.
• One-sided infinite limits are defined similarly:
  • $\lim<em>{x \to a^-} f(x) = \infty,\, \lim</em>{x \to a^+} f(x) = \infty,$
  • or for -∞ on the left/right, respectively.
• Vertical asymptotes:
  • The vertical line x = a is a vertical asymptote of y = f(x) if at least one of the following holds:
  • $\lim<em>{x \to a^-} f(x) = \infty \,\text{ or } \,\lim</em>{x \to a^-} f(x) = -\infty,$
  • $\lim<em>{x \to a^+} f(x) = \infty \,\text{ or } \,\lim</em>{x \to a^+} f(x) = -\infty.$
• Example: tan x
  • tan x = sin x / cos x, so potential vertical asymptotes occur where cos x = 0, i.e., at
  • $x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}.$
  • Near x = \frac{\pi}{2}, the function behaves as follows:
  • $\lim_{x \to (\pi/2)^-} \tan x = +\infty,$
  • $\lim_{x \to (\pi/2)^+} \tan x = -\infty.$
  • Thus x = \tfrac{\pi}{2} + k\pi are vertical asymptotes; the graph confirms this (Figure 14).
• Additional notes on tan x:
  • The graph near each asymptote shows the symmetric blow-up on either side.
  • The expression $\tan x = \frac{\sin x}{\cos x}$ helps explain why vertical asymptotes occur where cos x = 0.

Example 8 – Solution (Vertical Asymptotes of tan x)

• Potential vertical asymptotes occur at x = \tfrac{\pi}{2} + k\pi, for all integers k.
• The left-hand and right-hand limits confirm the asymptotic behavior:
  • As x → (\tfrac{\pi}{2})^-: tan x → +∞.
  • As x → (\tfrac{\pi}{2})^+: tan x → -∞.
• The graph in Figure 14 supports these conclusions.

Summary of Key Points

• A limit describes the value a function approaches as x gets arbitrarily close to a, ignoring the function value at a itself.
• Limits can be approached from either side; one-sided limits may exist even if the two-sided limit does not.
• If left and right limits exist and are equal, the two-sided limit exists and equals that common value.
• Some limits do not exist because the function oscillates without approaching a single value or because left/right limits diverge to infinity.
• Infinite limits and vertical asymptotes describe how a function behaves near points where it grows without bound; vertical asymptotes are lines x = a where the function becomes unbounded as x approaches a from either side.
• Practical examples include limits of sin x / x, limits of sin(p/x), and tan x near its asymptotes.