Calculus I · Limits and Continuity · lesson
Direct Substitution for Limits
Learn Direct Substitution for Limits with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
Where this chapter fits
Chapter 2: Finite limits and algebra
Turn indeterminate forms into solvable expressions using substitution, limit laws, factoring, conjugates, and piecewise reasoning.
Reading lens: What did direct substitution reveal, and which algebraic move removes the obstacle without changing nearby behavior? Keep that question in view while reading Direct Substitution for Limits; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects How to Read Limits From a Graph to Limit Laws and How to Use Them. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Evaluate limits of familiar continuous functions by substitution; interpret the result of substitution before choosing a more complicated method.
Evaluating Finite Limits
Start With Direct Substitution
Once you understand what a limit means, most homework problems become a method-selection problem. The first move should almost always be the simplest one:
First Move for Nearly Every Limit
Substitute the target input into the expression.
• If you get an ordinary real number, that is usually the limit. • If you get , simplify the expression and try again. • If you get a nonzero number divided by zero, analyze one-sided infinite behavior. • If the input approaches infinity and you get an indeterminate form such as , compare dominant terms.
Why does substitution work for so many functions? Because polynomials, rational functions away from zero denominators, root functions on their domains, and trigonometric functions on their domains are continuous. Continuity will be developed carefully in Chapter 5. For now, think of a continuous graph as one with no break at the target point.
Substitute and stop
Evaluate
Show worked solution
Replace by :
Nothing breaks, no denominator becomes zero, and no special method is needed. Therefore,
direct-sub-01Evaluate .
Your work stays on this device. No account or AI grader is used.
Show hint
Substitute because the linear function is continuous.
Attempt once to unlock the solution
Submit an answer first. The hint is available now.
Polynomial substitution
Evaluate
Show worked solution
Polynomials are continuous everywhere, so substitute :
Rational function with a safe denominator
Evaluate
Show worked solution
Substitute first:
The denominator approaches , not zero, so the quotient law is valid.
A root function
Evaluate
Show worked solution
The expression under the root approaches
Therefore,
Students sometimes overcomplicate a limit because the chapter is about limits. If substitution gives a legal real number, do not factor six polynomials, draw a sign chart, or summon l'Hospital's Rule from a future chapter. Stop.
Source & rights
Original instruction with traceable references.
The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.
The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.