Limits & Intro to Calculus: What a Function Approaches
Understand limits — what a function approaches near a point — plus the calculus ideas of instantaneous rate and slope of a curve, at the level the ACT and early calculus expect.
The Short Version
- A limit is the value a function approaches as x nears a point — even if the function never reaches it there.
- For most functions, evaluate a limit by direct substitution.
- When substitution gives 0/0, factor and cancel — often a removable hole.
- Limits underlie the calculus ideas of instantaneous rate and the slope of a curve. An ACT and Pre-Calc topic.
Calculus begins with one deceptively simple question: as x gets closer and closer to some number, what value does the function approach? That target value is the limit. The subtle part is that the function doesn't have to actually equal the limit at that point — it might have a hole there — yet it can still clearly be heading toward a specific value. Grasp that, and you've grasped the foundation of all of calculus.
This guide builds intuition for limits, shows how to evaluate the common cases, and connects them to calculus, with worked and practice problems matched to the level seen on the ACT and in early calculus at Northside Tutoring.
Why Limits Matter
The ACT occasionally includes limit and basic-calculus questions, and limits are the very first topic of any calculus course. Understanding them early gives students a real head start. They're a Pre-Calculus / introductory-calculus topic, beyond the SAT and SSAT.
The Core Idea
The notation limx→a f(x) = L reads "as x approaches a, f(x) approaches L." The emphasis is on approaching: we care about the function's behavior near a, not necessarily its value exactly at a.
Seeing a Limit
Even though the function has a hole at x = 2, it clearly approaches 3 from both sides — so the limit is 3.
Evaluating Simple Limits
For most "nice" functions (polynomials, and rationals where the denominator isn't zero), you find the limit by direct substitution — just plug in the value:
Holes & Removable Cases
Substitution sometimes gives the indeterminate form 0/0. That usually signals a removable hole: factor the numerator and denominator, cancel the common factor, then substitute:
0/0 means "factor," not "undefined"
Getting 0/0 doesn't mean the limit doesn't exist — it means you have more work to do. Factor and cancel to remove the hole, then substitute. The limit often exists even though the function itself is undefined at that exact point.
Why Limits Lead to Calculus
Limits are the tool that makes calculus possible. The derivative — the instantaneous rate of change, or the slope of a curve at a single point — is defined as the limit of average slopes as the interval shrinks to zero. So the "what is it approaching?" question becomes "how fast is it changing right here?" Every idea in calculus traces back to a limit.
Where You'll See This — Test by Test
Limits are a Pre-Calculus / introductory-calculus topic that surfaces on the ACT but not the SAT or SSAT. No reference sheet covers them — understanding the idea is what counts.
ACT
Occasionally tests limits and basic-calculus ideas like rate of change; the SAT does not.
Explore ACT Tutoring → College AdmissionsSAT
Not on the SAT — limits are beyond its scope. SAT prep focuses on algebra and data instead.
Explore SAT Tutoring → K-12 CurriculumPre-Calculus
The opening topic of Pre-Calculus and Calculus; mastering it early pays off.
Explore Math Tutoring → K-12 CurriculumAlgebra II
Builds on the function and factoring skills from Algebra II.
Explore Algebra Tutoring →Watch the Lesson
Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.
Limits — In Plain English
A live walkthrough from our tutoring team.
— Featuring a Northside Tutoring instructor
Worked Example Problems
These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.
Evaluate lim as x→4 of (2x + 1).
Show solution
Direct substitution: 2(4) + 1 = 9.
Evaluate lim as x→2 of (x² − 4)/(x − 2).
Show solution
Substitution gives 0/0. Factor: (x+2)(x−2)/(x−2) = x + 2. Then substitute: 2 + 2 = 4.
From a graph, a function approaches 5 from the left and 5 from the right at x = 1, but f(1) = 2 (a single plotted point). What is the limit as x→1?
Show solution
The limit depends on what the function approaches, not its value at the point: both sides approach 5, so the limit is 5.
Evaluate lim as x→0 of (x² + 3x)/x.
Show solution
Factor x: x(x + 3)/x = x + 3. Substitute 0: 0 + 3 = 3.
What does the derivative of a function represent geometrically?
Show solution
The slope of the curve at a single point — the instantaneous rate of change, defined as a limit of average slopes.
Common Mistakes to Avoid
Three traps that catch students every year
- Thinking 0/0 means no limit. It signals a removable hole — factor and cancel, then substitute.
- Using f(a) instead of the limit. A limit is about what the function approaches near a, which can differ from its value at a.
- Forgetting both sides must agree. If the function approaches different values from the left and right, the (two-sided) limit doesn't exist.
Practice Problems — You Try
Three problems below. Work each before checking the solution.
Evaluate lim as x→5 of (x² − 25)/(x − 5).
Show solution
Factor: (x+5)(x−5)/(x−5) = x + 5. Substitute: 5 + 5 = 10.
Evaluate lim as x→1 of (3x² − 2).
Show solution
Direct substitution: 3(1) − 2 = 1.
Evaluate lim as x→3 of (x² − 9)/(x² − x − 6).
Show solution
Factor both: (x+3)(x−3) / (x−3)(x+2) = (x+3)/(x+2). Substitute 3: 6/5.
The Northside Method — How We Teach This 1-on-1
Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.
Every Northside student works through a four-step framework:
- Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
- Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
- Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
- Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.
And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.
Ready to Turn This Concept Into Points?
Join a Northside cohort. Small-group instruction with our elite tutors, structured around your student's exact test or subject. Backed by our guarantee: hit your target, or earn 5 additional hours of cohort learning at no cost.
Online nationwide · In-person within 10 miles of Atlanta · Average SAT gain: 120+ points
Ready to begin?
Start tutoring with Northside.