Limits, direct substitution, and the basic limit laws

Key idea

A limit describes what value a function approaches as the input approaches a number: \[ \lim_{x\to a} f(x). \] If \(f\) is continuous at \(a\), then the limit is found by direct substitution: \[ \lim_{x\to a} f(x)=f(a). \] This works for polynomials, and also for rational functions as long as the denominator is not zero at \(a\).

Limit laws you’ll use constantly

• Sum: \(\lim (f+g)=\lim f+\lim g\)
• Product: \(\lim (fg)=(\lim f)(\lim g)\)
• Quotient: \(\lim \frac{f}{g}=\frac{\lim f}{\lim g}\) if \lim g≠ 0
• Constant multiple: \(\lim (cf)=c\lim f\)

Worked example

Try it

Summary

• For continuous functions, \(\lim_{x\to a} f(x)=f(a)\).
• Limit laws let you break complicated limits into simpler ones.

Indeterminate forms \(0/0\): factoring, canceling, and rationalizing

Key idea

If direct substitution produces an indeterminate form like \(\frac{0}{0}\), the limit is not found yet. Instead, simplify the expression (without plugging in) and then take the limit. Two common tools:

• Factor and cancel: factor the numerator/denominator and cancel a common factor (after factoring).
• Rationalize: multiply by a conjugate to remove radicals like \(\sqrt{x^2+1}-x\).

Worked example

Try it

Summary

• If you see \(0/0\), simplify (factor/cancel or rationalize) before substituting.
• After simplification, use direct substitution or limit laws.

Special limits: \(\dfrac{\sin x}{x}\), exponential limits, and scaling

Key idea

Two special limits appear everywhere in Calculus:

• \(\displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1\) (angles in radians)
• \(\displaystyle \lim_{x\to 0}\frac{e^x-1}{x}=1\)

You can combine these with substitutions. For example, if you see \(\sin(5x)\), let \(u=5x\) so that \(u\to 0\) when \(x\to 0\).

Worked example

Try it

Summary

• \(\lim_{x\to 0}\dfrac{\sin x}{x}=1\) and \(\lim_{x\to 0}\dfrac{e^x-1}{x}=1\).
• Use substitutions and scaling to match the special-limit forms.

Limits at infinity for rational functions and asymptotes

Key idea

For rational functions \(\dfrac{P(x)}{Q(x)}\), compare the degrees (highest powers) of \(P\) and \(Q\). A quick rule for \(\lim_{x\to\infty}\dfrac{P(x)}{Q(x)}\):

• If \(\deg(P) < \deg(Q)\), the limit is \(0\).
• If \(\deg(P) = \deg(Q)\), the limit is the ratio of leading coefficients.
• If \(\deg(P) > \deg(Q)\), the expression grows without bound (often \(\pm\infty\)), so there is no horizontal asymptote.

Worked example

Try it

Summary

• At infinity, compare degrees: smaller degree on top \(\Rightarrow 0\); equal degrees \(\Rightarrow\) ratio of leading coefficients.
• These limits often tell you the horizontal asymptote \(y=L\).

Continuity at a point and continuity of piecewise functions

Key idea

A function \(f\) is continuous at \(x=a\) if all three conditions hold:

• \(f(a)\) is defined,
• \(\lim_{x\to a} f(x)\) exists,
• \(\lim_{x\to a} f(x)=f(a)\).

The two-sided limit \(\lim_{x\to a} f(x)\) exists exactly when the one-sided limits match: \[ \lim_{x\to a^-} f(x)=\lim_{x\to a^+} f(x). \]

Worked example

Try it

Summary

• Continuity at \(a\): \(f(a)\) exists, \(\lim_{x\to a}f(x)\) exists, and they are equal.
• For piecewise functions, check left and right limits at the break point.

When limits fail: jump discontinuities, infinite limits, and DNE

Key idea

A two-sided limit \(\lim_{x\to a} f(x)\) exists only if the left and right limits are equal. If they are different, the limit does not exist (often a jump discontinuity). If the function grows without bound (toward \(\pm\infty\)), you have an infinite discontinuity (a vertical asymptote).

Worked example

Try it

Summary

• If left ≠ right, the two-sided limit does not exist (jump-type behavior).
• If the function blows up to \(\pm\infty\), you have an infinite limit and a vertical asymptote.

Why limits and continuity matter

Where limits and continuity show up

• Derivatives: the derivative is defined using a limit of a difference quotient.
• Integrals: area and accumulation are built from limits of sums.
• Graphs: continuity explains when a graph can be drawn without lifting your pencil.
• Modeling: physics, economics, and biology use limits to describe “instantaneous” behavior and long-run trends.

Worked example: filling a hole to make a function continuous

Try it

Final recap

• Direct substitution: if \(f\) is continuous at \(a\), then \(\lim_{x\to a} f(x)=f(a)\).
• 0/0 limits: simplify first (factor/cancel or rationalize), then substitute.
• Special limits: \(\lim_{x\to 0}\dfrac{\sin x}{x}=1\) and \(\lim_{x\to 0}\dfrac{e^x-1}{x}=1\).
• Limits at infinity: compare degrees or divide by the highest power of \(x\).
• One-sided limits: a two-sided limit exists only if left and right match.
• Continuity: \(f(a)\) defined, limit exists, and \(\lim_{x\to a} f(x)=f(a)\).