Suggestions
Use up and down arrows to review and enter to select.Please wait while we process your payment
If you don't see it, please check your spam folder. Sometimes it can end up there.
If you don't see it, please check your spam folder. Sometimes it can end up there.
Please wait while we process your payment
By signing up you agree to our terms and privacy policy.
Don’t have an account? Subscribe now
Create Your Account
Sign up for your FREE 7-day trial
Already have an account? Log in
Your Email
Choose Your Plan
Individual
Group Discount
Save over 50% with a SparkNotes PLUS Annual Plan!
Purchasing SparkNotes PLUS for a group?
Get Annual Plans at a discount when you buy 2 or more!
Price
$24.99 $18.74 /subscription + tax
Subtotal $37.48 + tax
Save 25% on 2-49 accounts
Save 30% on 50-99 accounts
Your Plan
Payment Details
Payment Summary
SparkNotes Plus
You'll be billed after your free trial ends.
7-Day Free Trial
Not Applicable
Renews March 26, 2024 March 19, 2024
Discounts (applied to next billing)
DUE NOW
US $0.00
SNPLUSROCKS20 | 20% Discount
This is not a valid promo code.
Discount Code (one code per order)
SparkNotes PLUS Annual Plan - Group Discount
Qty: 00
SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.
Choose Your Plan
For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!
You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.
Members will be prompted to log in or create an account to redeem their group membership.
Thanks for creating a SparkNotes account! Continue to start your free trial.
Please wait while we process your payment
Your PLUS subscription has expired
Please wait while we process your payment
Please wait while we process your payment
An asymptote is a line that a graph approaches without touching.
If a graph has a horizontal asymptote of y = k, then part of the graph
approaches the line y = k without touching it--y is almost equal to k, but
y is never exactly equal to k. The following graph has a horizontal
asymptote of y = 3:
The following graph has a horizontal asymptote of y = 0:
If a graph has a vertical asymptote of x = h, then part of the graph
approaches the line x = h without touching it--x is almost equal to h,
but x is never exactly equal to h. The following graph has a vertical
asymptote of x = 3:
One reason vertical asymptotes occur is due to a zero in the denominator of a
rational function. For example, if f (x) = , then x cannot
equal 5, but x can equal values very close to 5 (4.99, for example). The
graph of f (x) = looks like:
Similarly, horizontal asymptotes occur because y can come close to a value, but can never equal that value. In the previous graph, there is no value of x for which y = 0 (≠ 0), but as x gets very large or very small, y comes close to 0. Thus, f (x) = has a horizontal asymptote at y = 0.
The graph of a function may have several vertical asymptotes. f (x) = has vertical asymptotes of x = 2 and x = - 3, and f (x) = has vertical asymptotes of x = - 4 and x = . In general, a vertical asymptote occurs in a rational function at any value of x for which the denominator is equal to 0, but for which the numerator is not equal to 0.
Please wait while we process your payment