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Calculus - Limits

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    Study Guide Calculus – Limits 1. Evaluating Limits 1. What Does It Mean to Evaluate a Limit? A limit describes the value that a function or sequence approaches as the input gets closer and closer to a certain number. Limits can be evaluated using several methods, including: • Recognizing patterns • Direct substitution • Algebraic simplification (factoring, canceling terms, etc.) Which method you use depends on the type of expression involved. Example 1 : Limit of a Sequence Consider the sequence: Each fraction is slightly larger than the one before it. In every term, the numerator is one less than the denominator , so the value of each fraction gets closer and closer to 1. Therefore, the limit of the sequence is: Example 2 : Limit by Direct Substitution Evaluate:

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    Study Guide This expression is defined at (x = 2), so we can substitute directly: Example 3 : Limit Requiring Factoring Evaluate: Substituting (x = - 3) gives: which is undefined . This means we must simplify first. Factor the numerator: Now simplify: Now take the limit: Graphical Interpretation

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    Study Guide Figure 1 The graph of y = (x 2 − 9)/(x + 3). The graph of is the same as the graph of the line: except for one missing point at (( - 3, - 6)). Even though the function is not defined at (x = - 3), the limit still exists and equals – 6 . Example 4 : Simplifying a Compound Fraction Evaluate: Substituting (x = 3) gives (0/0), so we simplify.

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    Study Guide After simplifying the compound fraction and canceling common factors, the expression reduces to: Now substitute: Example 5 : Limit That Equals Zero Evaluate: Substitute (x = 0): Example 6 : Limit That Does Not Exist Evaluate: Substituting (x = 0) gives: which is undefined. The expression grows without bound as (x) approaches 0, so the limit does not exist (DNE) . (Remember: infinity is not a real number , so it cannot be the value of a limit.)

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    Study Guide Key Takeaways • A limit describes what a value approaches , not necessarily what it equals • Try direct substitution first • If you get (0/0), simplify algebraically • A limit can exist even if the function is undefined at a point • Division by zero means the limit does not exist 2. One - Sided Limits 1. What Are One - Sided Limits? Sometimes, it is not enough to look at how a function behaves near a point from both sides . Instead, we may need to consider the behavior from only one direction . • If (x) approaches a value (c) from the right , we write: • If (x) approaches (c) from the left , we write: 2. When Does a Two - Sided Limit Exist? A two - sided limit exists only if both one - sided limits exist and are equal . That is, if and only if

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    Study Guide If the left - hand and right - hand limits are different — or if one does not exist — then the limit does not exist (DNE) . Example 1 : Right - Hand Limit Evaluate: As (x) approaches 0 from the right , (x) is always nonnegative. The square root of (x) gets closer and closer to 0. Therefore: In this case, direct substitution also gives the correct answer, but that is not always true , as the next example shows. Example 2 : Left - Hand Limit Evaluate: As (x) approaches 0 from the left , (x) is negative. The square root of a negative number is not defined in the real numbers. So: Since: the two - sided limit

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    Study Guide also does not exist . Example 3 : One - Sided Limits with Absolute Value Evaluate the following limits: (a) Left - hand limit As (x → 2 - ), the expression (x - 2) is negative , so: Thus: So: (b) Right - hand limit As (x → 2 + ), (x - 2) is positive , so: Thus:

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    Study Guide So: (c) Two - sided limit Because: the two - sided limit: does not exist (DNE) . Key Takeaways • One - sided limits look at behavior from only one direction • (x → c - ) means from the left • (x → c + ) means from the right • A two - sided limit exists only if both one - sided limits are equal • If left and right limits differ, the limit is DNE • Absolute value expressions often require checking one - sided limits 3. Infinite Limits 1. What Is an Infinite Limit? Sometimes, as the input value (x) gets close to a certain number, a function does not approach a finite value.

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    Study Guide Instead, the function may: • Increase without bound (go upward forever), or • Decrease without bound (go downward forever) When this happens, the function is said to have an infinite limit . We write: 2. Vertical Asymptotes If a function has an infinite limit as (x) approaches a number (c), then the graph has a vertical asymptote at: This means the graph gets very close to the vertical line (x = c) but never touches it. 3. Infinite Limits in Rational Functions A rational function often has an infinite limit when: • The denominator approaches 0 • The numerator approaches a nonzero value The sign of the infinite limit depends on: • The sign of the numerator • The sign of the denominator near the point being approached Example 1 : A Basic Infinite Limit Evaluate:

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    Study Guide As (x) approaches 0: • The numerator (1) is always positive • The denominator (x^2) approaches 0 and is always positive So the fraction increases without bound: Figure 1 The graph of y = 1/ x 2 . This figure shows that the graph rises sharply on both sides of (x = 0), confirming a vertical asymptote at (x = 0). Example 2 : One - Sided Infinite Limit Evaluate: As (x) approaches 2 from the left : • The numerator approaches (5) (positive) • The denominator approaches (0) through negative values

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