Calculating Limits Using Limit Laws

Ever wonder how to find limits without resorting to endless graphing? The Direct Substitution Property (DSP) is your best friend when evaluating limits algebraically.

The DSP states that if f(x) is a polynomial or rational function, and x = a is in the domain of f(x), then: $$\lim_{x \to a} f(x) = f(a)$$

This means you can simply plug the value into the function! But what if direct substitution gives you $\frac{0}{0}$ or another undefined expression?

When the DSP doesn't work immediately:

1. Try factoring to eliminate common terms
2. Expand expressions when helpful
3. Use conjugates for expressions with roots
4. Find common denominators if needed

⚠️ Remember: The limit of a function exists only if both the left-hand limit and right-hand limit exist and equal the same value, and that value must be a real number (not infinity).

For example, to find $\lim_{x \to 5} (7x - 2)$, we can directly substitute: $= 7(5) - 2 = 33$

But when evaluating $\lim_{t \to \frac{\pi}{2}} \tan(t)$, we can't use the DSP since $\frac{\pi}{2}$ isn't in the domain of tangent. In this case, using a graph or table shows that this limit doesn't exist.

Working with Indeterminate Forms

When limits give you a headache by resulting in $\frac{0}{0}$ at first glance, you need special algebraic techniques to uncover their true values.

For example, when finding $\lim_{x \to 3} \frac{x^2-9}{x-3}$, direct substitution gives $\frac{0}{0}$ (indeterminate). But factoring helps: $$\lim_{x \to 3} \frac{x^2-9}{x-3} = \lim_{x \to 3} \frac{$x-3$$x+3$}{x-3} = \lim_{x \to 3}$x+3$ = 6$$

For expressions involving roots, the conjugate method is powerful. Consider finding $\lim_{t \to 0} \frac{3-\sqrt{t+9}}{t}$:

First try direct substitution: $\frac{3-3}{0}$ (indeterminate) Multiply by the conjugate: $\lim_{t \to 0} \frac{(3-\sqrt{t+9})(3+\sqrt{t+9})}{t(3+\sqrt{t+9})}$ Simplify: $\lim_{t \to 0} \frac{9-(t+9)}{t(3+\sqrt{t+9})} = \lim_{t \to 0} \frac{-t}{t(3+\sqrt{t+9})} = \lim_{t \to 0} \frac{-1}{3+\sqrt{t+9}} = \frac{-1}{6}$

🔑 Pro Tip: Don't drop the "lim" notation until you reach a point where you can apply the DSP. Keeping track of the limit operation helps avoid errors in your calculations.

Remember your checklist:

1. Try DSP first
2. If you get $\frac{0}{0}$, try factoring
3. If the function isn't defined at the point, use a graph or table
4. For roots, consider using the conjugate method

Special Techniques for Complex Limits

When tackling limits with multiple terms or roots, expanding expressions can reveal simpler forms that are easier to evaluate.

For example, to find $\lim_{h \to 0} \frac{(11+h)^2 - 121}{h}$, start by expanding the numerator: $$\lim_{h \to 0} \frac{121+22h+h^2 - 121}{h} = \lim_{h \to 0} \frac{22h+h^2}{h} = \lim_{h \to 0} $22+h$ = 22$$

The conjugate method is especially useful for limits involving radicals. Consider $\lim_{t \to 0} (3 - \sqrt{t+9})$:

Multiply by $\frac{3 + \sqrt{t+9}}{3 + \sqrt{t+9}}$ to rationalize: $$\lim_{t \to 0} \frac{$3 - \sqrt{t+9}$$3 + \sqrt{t+9}$}{3 + \sqrt{t+9}} = \lim_{t \to 0} \frac{9 - $t+9$}{3 + \sqrt{t+9}} = \lim_{t \to 0} \frac{-t}{3 + \sqrt{t+9}} = \frac{0}{6} = 0$$

For reciprocal expressions like $\lim_{h \to 0} \frac{(3+h)^{-1} - 3^{-1}}{h}$, rewrite using common fractions: $$\lim_{h \to 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h} = \lim_{h \to 0} \frac{3 - $3+h$}{3h$3+h$} = \lim_{h \to 0} \frac{-h}{3h$3+h$} = \lim_{h \to 0} \frac{-1}{3$3+h$} = \frac{-1}{9}$$

💡 Remember: These techniques transform indeterminate forms into expressions where direct substitution works. The key is identifying which technique to apply based on the structure of the limit problem.

Simplifying Limits with Absolute Values

Absolute value expressions require special attention in limits, as their behavior changes depending on whether the input is positive or negative.

For expressions of the form $\frac{|f(x)|}{f(x)}$, the result depends on the sign of f(x):

• If f(x) > 0, then $\frac{|f(x)|}{f(x)} = 1$
• If f(x) < 0, then $\frac{|f(x)|}{f(x)} = -1$
• If f(x) = 0, the expression is undefined

This means that when evaluating limits involving $\frac{|f(x)|}{f(x)}$, you need to consider left and right-hand limits separately.

For example, when finding $\lim_{x \to 2} \frac{x^2 - 4}{|x - 2|}$:

First, simplify: $\frac{x^2 - 4}{|x - 2|} = \frac{(x-2)(x+2)}{|x-2|}$

For x < 2, we have $\frac{|x-2|}{x-2} = -1$, so: $\lim_{x \to 2^-} \frac{(x-2)(x+2)}{|x-2|} = \lim_{x \to 2^-} (x+2)(-1) = -4$

For x > 2, we have $\frac{|x-2|}{x-2} = 1$, so: $\lim_{x \to 2^+} \frac{(x-2)(x+2)}{|x-2|} = \lim_{x \to 2^+} (x+2) = 4$

Since the left and right limits are different, $\lim_{x \to 2} \frac{x^2 - 4}{|x - 2|}$ doesn't exist.

🔍 When working with absolute values in limits, always examine how the expression behaves on both sides of the point in question. The behavior often changes at the transition point.

The Squeeze Theorem: A Powerful Tool

When direct algebraic methods fail, the Squeeze Theorem offers an elegant way to find limits by "trapping" a function between two simpler functions.

The Squeeze Theorem states: If f(x) ≤ g(x) ≤ h(x) near a (except possibly at a) and both lim f(x) and lim h(x) equal L as x → a, then lim g(x) = L as well.

Think of it like this: if g(x) is sandwiched between two functions that both approach the same value, then g(x) must approach that value too!

For example, to find $\lim_{x \to 4} g(x)$ given that $-10x \leq g(x) \leq -6x - x^2$:

First, calculate the limits of the bounding functions: $\lim_{x \to 4} (-10x) = -40$ $\lim_{x \to 4} (-6x - x^2) = -24 - 16 = -40$

Since both bounds approach -40, by the Squeeze Theorem: $\lim_{x \to 4} g(x) = -40$

This is especially useful for oscillating functions. For instance, to find $\lim_{x \to 0} x^{10} \cos(\frac{5}{x})$:

We know $-1 \leq \cos(\frac{5}{x}) \leq 1$, so: $-x^{10} \leq x^{10}\cos(\frac{5}{x}) \leq x^{10}$

As $x \to 0$, both $-x^{10}$ and $x^{10}$ approach 0, so: $\lim_{x \to 0} x^{10}\cos(\frac{5}{x}) = 0$

✨ The Squeeze Theorem is your secret weapon for functions with complicated behavior. It's particularly useful for products involving trigonometric functions or other oscillating terms.

Understanding Function Continuity

A function f(x) is continuous at x = a if there's no hole, jump, or asymptote at that point. This requires three specific conditions:

1. The limit $\lim_{x \to a} f(x)$ exists
2. The function value f(a) exists (is defined)
3. The limit equals the function value: $\lim_{x \to a} f(x) = f(a)$

All three conditions must be true for a function to be continuous at a point. If even one fails, the function is discontinuous there.

For example, consider the piecewise function: $$f(t) = \begin{cases} \frac{t-1}{t^2-1} & \text{if } t \neq 1 \ 2 & \text{if } t = 1 \end{cases}$$

To check continuity at t = 1:

• The limit $\lim_{t \to 1} f(t) = \frac{1}{2}$ (verify this)
• The function value f(1) = 2 (given in the definition)
• Since $\frac{1}{2} \neq 2$, the function is discontinuous at t = 1

To find values that make a piecewise function continuous, equate the limits at transition points with the defined function values.

🔄 Continuity is crucial for many calculus theorems. A function that's continuous on an interval can be differentiated and integrated on that interval, and it satisfies the Intermediate Value Theorem.

The Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a powerful tool that guarantees the existence of solutions within continuous functions.

If f(x) is continuous on an interval $a,b$ and N is any number between f(a) and f(b), then there exists at least one value c in the interval $a,b$ where f(c) = N.

In simple terms: if a continuous function has outputs of y = 3 and y = 7, it must hit every value between 3 and 7 somewhere along the way.

This is especially useful for finding roots $where f(c) = 0$:

1. Find an interval $a,b$ where f(a) and f(b) have opposite signs
2. By the IVT, there must be a root somewhere in that interval

For example, to show that cos(x) = x has a solution in (0,1):

• Let f(x) = cos(x) - x
• f(0) = cos(0) - 0 = 1 > 0
• f(1) = cos(1) - 1 ≈ 0.54 - 1 = -0.46 < 0
• Since f(0) > 0 and f(1) < 0, by the IVT, there must be a value c in (0,1) where f(c) = 0

🎯 The IVT doesn't tell you exactly where the solution is—just that it exists. For finding the precise value, you'd need to use numerical methods like bisection or Newton's method.

Working with Limits at Infinity

Understanding what happens to functions as x gets extremely large (or extremely negative) is crucial for analyzing long-term behavior.

When we write $\lim_{x \to \infty} f(x) = L$, we mean that as x grows without bound in the positive direction, f(x) gets arbitrarily close to L.

For rational functions $\frac{P(x)}{Q(x)}$ as x approaches infinity:

1. Divide both numerator and denominator by the highest power of x in the denominator
2. As x → ∞, terms with x in the denominator approach 0

For example, for $\lim_{x \to \infty} \frac{5x^2 + 2x - 1}{4x^3 + 5x + 2}$:

Divide by x³ (highest power in denominator): $$\lim_{x \to \infty} \frac{5x^2 + 2x - 1}{4x^3 + 5x + 2} = \lim_{x \to \infty} \frac{\frac{5}{x} + \frac{2}{x^2} - \frac{1}{x^3}}{4 + \frac{5}{x^2} + \frac{2}{x^3}} = \frac{0}{4} = 0$$

The horizontal asymptote is the line y = L, where L is the limit as x approaches ±∞.

📈 Horizontal asymptotes tell you the "end behavior" of a function—what happens far to the left or right on the graph. A function can cross its horizontal asymptote, unlike vertical asymptotes.

Limits at Infinity and Horizontal Asymptotes

When x grows extremely large (positively or negatively), many functions settle toward a specific value—this value becomes a horizontal asymptote.

For a function to have a horizontal asymptote at y = L, either:

• $\lim_{x \to \infty} f(x) = L$ (right side asymptote)
• $\lim_{x \to -\infty} f(x) = L$ (left side asymptote)

A function can have different horizontal asymptotes on the left and right sides. For example, a function might approach y = 2 as x → ∞ and y = -1 as x → -∞.

When sketching graphs with horizontal asymptotes:

1. Plot the asymptote as a dashed horizontal line
2. Show the function approaching (but not necessarily touching) this line
3. Remember that functions can cross their horizontal asymptotes multiple times

For rational functions, the horizontal asymptote behavior depends on the degrees of the numerator and denominator polynomials:

• If degree of numerator < degree of denominator: y = 0 is the horizontal asymptote
• If degrees are equal: y = (leading coefficient of numerator)/(leading coefficient of denominator)
• If degree of numerator > degree of denominator: no horizontal asymptote (possibly a slant asymptote)

🌊 Think of horizontal asymptotes as the "eventual" behavior of a function. As x moves farther from the origin, the function values gradually settle toward their asymptotic value.

Calculating Important Limits at Infinity

Understanding how different types of functions behave as x approaches infinity gives you powerful tools for analyzing their end behavior.

Key limits to remember:

• $\lim_{x \to \infty} \frac{1}{x^r} = 0$ for any positive value of r
• $\lim_{x \to \infty} e^x = \infty$ and $\lim_{x \to -\infty} e^x = 0$
• $\lim_{x \to \pm\infty} \sin(x)$ doesn't exist (oscillates forever)

For rational functions, follow this process:

1. Identify the highest power of x in the denominator
2. Divide both numerator and denominator by this highest power
3. Apply the limit rules to simplify

For example, with $\lim_{x \to \infty} \frac{5x^2 + 2x - 1}{4x^3 + 5x + 2}$:

Divide everything by x³: $$\frac{5x^2 + 2x - 1}{4x^3 + 5x + 2} = \frac{\frac{5}{x} + \frac{2}{x^2} - \frac{1}{x^3}}{4 + \frac{5}{x^2} + \frac{2}{x^3}}$$

As x → ∞, all terms with x in the denominator approach 0: $$\lim_{x \to \infty} \frac{5x^2 + 2x - 1}{4x^3 + 5x + 2} = \frac{0}{4} = 0$$

💫 The key insight: terms with higher powers of x in the denominator approach zero faster. This allows us to determine which terms dominate as x gets very large.