Calculus (MATH 1500): Limit Laws

What happens when a fraction has a denominator approaching 0?

$$\frac{\left(+\right)}{tiny\left(+\right)}\to+\infty$$

$$\frac{\left(+\right)}{tiny\left(-\right)}\to-\infty$$

$$\frac{\left(-\right)}{tiny\left(+\right)}\to-\infty$$

$$\frac{\left(-\right)}{tiny\left(-\right)}\to+\infty$$

What happens when a fraction has a denominator approaching ꝏ?

$$\frac{k}{\infty}\to0$$

$$\frac{k}{-\infty}\to0$$

How do you determine if f(x) is continuous at x = a?

> Does f(a) exist?

• No → not continuous.

• Yes → continue.

> Does lim_x→a f(x) exist?

• No → not continuous.

• Yes → continue.

> Is lim_x→a f(x) = f(a)?

• Yes → continuous

• No → not continuous.

Why is a f(x) not continuous at a point if the limit doesn’t exist?

> Continuity = no gaps at that point

> If limit DNE, the left and right values don’t meet → there’s a gap → not continuous.

Why does lim f(x) ≠ f(a) as x→a mean not continuous?

> Continuity means the graph reaches the same point it approaches.

> If the approached value and actual value differ, there’s a break/gap.

How do you find the limits of rational functions at some constant?

> Substitute x = a.

> Is Q(a) ≠ 0?

• Yes → direct substitution → done.

• No → continue.

> Do you get 0/0?

• Yes → simplify → re-evaluate.

• No (nonzero/0) → check left/right limits.

> Do left/right limits agree?

• Yes → limit exists.

• No → DNE.

Note: If you see sign(x-a) → check left and right immediately.

How do you evaluate limits at ထ?

> Is the function rational?

• Yes → deg(N) < deg(D) = 0 ; deg(N) = deg(D) = ratio of leading coefficients ; deg(N) > deg(D) → keep only dominant terms → evaluate end behaviour.

• No → continue.

> Did you get ထ - ထ?

• Yes → rationalize using the conjugate → re-evaluate

• No → continue.

> Does the function contain radicals?

• Yes → factor highest power from inside the root → simplify → evaluate limit.

• No → keep only dominant terms → evaluate end behaviour.

How do you evaluate limits of piecewise functions?

> Find the left-hand limit and right-hand limit. Do the left-hand and right-hand limits exist?

• Yes → continue.

• No → DNE.

> Do the limits agree?

• Yes = L.

• No = DNE.

How do you find parameter values so a piecewise limit exists?

> Is each piece continuous on its own interval?

• Yes → continue.

• No → check discontinuities.

> Find LHL and RHL. Do the limits agree?

• Yes → set LHL = RHL → solve for the parameter.

How do you find parameter values so a piecewise function is continuous?

> Find the limit via computing LHL and RHL.

> Find the function value (use the piece that contains x = a).

> Apply the continuity condition f(a) = lim_x→a f(x).

> Substitute and solve for the parameter.

> Recalculate the RHL and/or LHL with the new knowns and evaluate the limit. (this is the parameter’s value so that the function is equalling said value).

How do you evaluate limits from a graph?

> Are you finding the left- or right-hand limit?

• Yes → Follow the graph from the left/right toward x = a → Read the y-value being approached.

• No (you are finding lim_x→a f(x)) → continue.

> Find the LHL and RHL → agree (L) / disagree (DNE).

When should you use the Squeeze Theorem?

Look for sin(something) or cos(something) that is multiplied by a factor approaching 0.

How do I solve a limit using the Squeeze Theorem?

> Find bounds.

$$-1\le\sin\left(\theta\right)\le1$$

$$-1\le\cos\left(\theta\right)\le1$$

> Multiply the inequality by any positive factor outside the trig function.

> Take limits of the outer functions. Do both outer limits equal the same value L?

• Yes → lim f(x) = L.

• No → Squeeze Theorem cannot be used.

NOTE: Add constant after.

What is a vertical asymptote?

A line x = a where the function approaches ±ꝏ.

How do you check for vertical asymptotes in a non-rational function?

Check whether the function approaches ±ꝏ at a finite x-value.

How do you check for vertical asymptotes in a rational function?

> Factor the fraction completely.

> Cancel common factors.

• Cancels → hole.

• Doesn’t cancel → potential candidate.

> Set Q(x) = 0 to find candidates.

> Make a sign chart of all remaining factors and their approaching values. If at least one one-sided limit is +ထ or -ထ, then x = a is a VA.

> Determine signs of f(x) on the left and right.

> Since denominator → 0, convert sign into +ထ or -ထ.

> State the one-sided limits and conclude VA.

What is a horizontal asymptote?

A line y = L that the function approaches as x approaches ꝏ and/or x approaches -ꝏ.

How do you check for horizontal asymptotes?

> Is f(x) a rational function?

• Yes → deg(N) < deg(D) → y = 0 ; deg(N) = deg(D) → ratio of leading coefficients ; deg(N) > deg(D) → no HA

• No → continue.

> Keep dominant terms → simplify → evaluate lim_x→-ꝏ and _x→+ꝏ. Do the limits equal?

• Yes/same value L → y = L

• No/different values L₁ and L₂ = y = L₁ and L₂

• Not finite → no HA.

How do you evaluate trig limits?

> Plug in x = a. Do you get a real number?

• Yes → evaluate directly.

• No → continue.

> Do you get a standard trig form?

• Yes → Rewrite to match the identity and evaluate.

• No → continue.

>Do you get 0/0?

• Yes → simplify and re-evaluate.

• No → continue.

> Do you see bounded oscillation?

• Yes → use Squeeze Theorem.

• No → continue.

> Does the expression blow up?

• Check one-sided limits.If LHL = RHL → limit exists. Otherwise → DNE.

How do you show that an equation has a solution in an interval?

> Define the function.

> Check continuity.

> Evaluate endpoints f(a) and f(b).

> Check sign change.

> Apply IVT: Since f(x) is cont. on [INTERVAL], and 0 lies between f(a) and f(b), the IVT guarantees that there is some number c in (INTERVAL) such that: f(c) = 0. Final answer: equation = 0 has at least one solution in the interval.

How do you show that two graphs intersect?

> Define the two functions.

> Rewrite as one function.

> Check continuity of each function.

> Evaluate two convenient x-values (e.g. 0 and 1) as endpoints.

> Check sign change.

> Apply IVT: Since [graph] and [graph] are continuous, f(x) = [EQUATION] is continuous on [INTERVAL], and since f(a) < 0 < f(b), there exists c in (0,1), such that f(c) = 0. Thus, [GRAPH] - [GRAPH] = 0, Therefore, the graphs intersect.

What are the trig limits you must memorize?

$$\lim_{x\to0}\frac{\sin x}{x}=1$$

$$\lim_{x\to0}\frac{x}{\sin x}=1$$

$$\lim_{x\to0}\frac{\tan x}{x}=1$$

$$\lim_{x\to0}\frac{x}{\tan x}=1$$

$$\lim_{x\to0}\frac{1-\cos x}{x}=0$$

$$\lim_{x\to0}\frac{\cos x-1}{x}=0$$