UNIT 6 • CALCULUS

Limit Calculator
Substitution, Factoring & L'Hôpital

A limit is the value $f(x)$ approaches as $x$ approaches a point. Try substituting first; if that gives $\frac{0}{0}$ , factor and cancel, or use L'Hôpital's rule.

THE ORDER TO TRY0/0 · ∞/∞

\begin{gathered} \lim_{x \to a} f(x) = f(a) \ \text{(if defined)} \\ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \ \text{(for } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}\text{)} \end{gathered}

1. SubstitutePut x = a in. A normal number is the answer.

2. Simplify 0/0Factor and cancel, or rationalise a square root, then substitute again.

3. L'HôpitalStill 0/0 or ∞/∞? Differentiate top and bottom separately and try again.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find $\lim_{x \to 2} \frac{x^2 - 5x + 6}{x^2 - 4}$ .

STEP 01 • TRY SUBSTITUTING

\frac{2^2 - 5(2) + 6}{2^2 - 4} = \frac{0}{0}

0/0 is indeterminate: it doesn't tell us the limit, so simplify first.

STEP 02 • FACTOR AND CANCEL

Both top and bottom are 0 at x = 2, so both have the factor $(x - 2)$ :

\frac{x^2 - 5x + 6}{x^2 - 4} = \frac{(x - 2)(x - 3)}{(x - 2)(x + 2)} = \frac{x - 3}{x + 2} \quad (x \neq 2)

STEP 03 • SUBSTITUTE AGAIN

\lim_{x \to 2} \frac{x - 3}{x + 2} = \frac{2 - 3}{2 + 2} = -\frac{1}{4}

FINAL CONCLUSION

The limit is −1/4

\lim_{x \to 2} \frac{x^2 - 5x + 6}{x^2 - 4} = -\frac{1}{4}

The calculator below opens on this limit. Tap sin(x)/x to see L'Hôpital's rule instead.

LABORATORY

Interactive Limit Calculator

LABORATORY

Limit Engine

f(x) =as x →

Try:

Approach from

Type inf or -inf for infinity. Use ^ for powers and brackets for functions: sin(x), ln(x), sqrt(x).

Common Questions

What is a limit?

The value a function gets closer and closer to as x approaches a point, even if the function is not defined at that point.

\frac{x^2 - 4}{x - 2}

has no value at x = 2, but it gets as close as you like to 4.

When can you use L'Hôpital's rule?

Only for the forms

\frac{0}{0}

and

\frac{\infty}{\infty}

. Then

\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}

: differentiate the top and the bottom separately (not with the quotient rule). Repeat if the form is still 0/0 or ∞/∞.

Why is the limit of sin x / x equal to 1?

It is a standard limit, true when x is in radians: near 0 the curve

\sin x

and the line

y = x

are almost the same. L'Hôpital's rule confirms it:

\lim_{x\to0} \frac{\cos x}{1} = 1

When does a limit not exist?

When the left and right limits are different (like

\frac{|x|}{x}

at 0, which gives −1 and 1), when the values grow without bound in opposite directions (like

\frac{1}{x}

at 0), or when they never settle (like

\sin\frac{1}{x}

at 0).

PreviousChi-Square Test

Unit 6•45 of 63

NextDerivative Calculator

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 6 • CALCULUS

Limit Calculator
Substitution, Factoring & L'Hôpital

A limit is the value $f(x)$ approaches as $x$ approaches a point. Try substituting first; if that gives $\frac{0}{0}$ , factor and cancel, or use L'Hôpital's rule.

THE ORDER TO TRY0/0 · ∞/∞

\begin{gathered} \lim_{x \to a} f(x) = f(a) \ \text{(if defined)} \\ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \ \text{(for } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}\text{)} \end{gathered}

1. SubstitutePut x = a in. A normal number is the answer.

2. Simplify 0/0Factor and cancel, or rationalise a square root, then substitute again.

3. L'HôpitalStill 0/0 or ∞/∞? Differentiate top and bottom separately and try again.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find $\lim_{x \to 2} \frac{x^2 - 5x + 6}{x^2 - 4}$ .

STEP 01 • TRY SUBSTITUTING

\frac{2^2 - 5(2) + 6}{2^2 - 4} = \frac{0}{0}

0/0 is indeterminate: it doesn't tell us the limit, so simplify first.

STEP 02 • FACTOR AND CANCEL

Both top and bottom are 0 at x = 2, so both have the factor $(x - 2)$ :

\frac{x^2 - 5x + 6}{x^2 - 4} = \frac{(x - 2)(x - 3)}{(x - 2)(x + 2)} = \frac{x - 3}{x + 2} \quad (x \neq 2)

STEP 03 • SUBSTITUTE AGAIN

\lim_{x \to 2} \frac{x - 3}{x + 2} = \frac{2 - 3}{2 + 2} = -\frac{1}{4}

FINAL CONCLUSION

The limit is −1/4

\lim_{x \to 2} \frac{x^2 - 5x + 6}{x^2 - 4} = -\frac{1}{4}

The calculator below opens on this limit. Tap sin(x)/x to see L'Hôpital's rule instead.

LABORATORY

Interactive Limit Calculator

LABORATORY

Limit Engine

f(x) =as x →

Try:

Approach from

Type inf or -inf for infinity. Use ^ for powers and brackets for functions: sin(x), ln(x), sqrt(x).

Common Questions

What is a limit?

The value a function gets closer and closer to as x approaches a point, even if the function is not defined at that point.

\frac{x^2 - 4}{x - 2}

has no value at x = 2, but it gets as close as you like to 4.

When can you use L'Hôpital's rule?

Only for the forms

\frac{0}{0}

and

\frac{\infty}{\infty}

. Then

\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}

: differentiate the top and the bottom separately (not with the quotient rule). Repeat if the form is still 0/0 or ∞/∞.

Why is the limit of sin x / x equal to 1?

It is a standard limit, true when x is in radians: near 0 the curve

\sin x

and the line

y = x

are almost the same. L'Hôpital's rule confirms it:

\lim_{x\to0} \frac{\cos x}{1} = 1

When does a limit not exist?

When the left and right limits are different (like

\frac{|x|}{x}

at 0, which gives −1 and 1), when the values grow without bound in opposite directions (like

\frac{1}{x}

at 0), or when they never settle (like

\sin\frac{1}{x}

at 0).

PreviousChi-Square Test

Unit 6•45 of 63

NextDerivative Calculator

Limit Calculator Substitution, Factoring & L'Hôpital

Step-by-Step Worked Example

The limit is −1/4

Interactive Limit Calculator

Limit Engine

Common Questions

NETZ Engine

Limit Calculator Substitution, Factoring & L'Hôpital

Step-by-Step Worked Example

The limit is −1/4

Interactive Limit Calculator

Limit Engine

Common Questions

Limit Calculator
Substitution, Factoring & L'Hôpital

Limit Calculator
Substitution, Factoring & L'Hôpital