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Calculus - Limits

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Page 1 of 33
Calculus - Limits - Page 1 preview imageStudy GuideCalculus–Limits1.Evaluating Limits1. What Does It Mean to Evaluate a Limit?Alimitdescribes the value that a function or sequenceapproachesas the input gets closer andcloser 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.Example1:Limit of a SequenceConsider the sequence:Each fraction is slightly larger than the one before it.In every term, the numerator isone less than the denominator, so the value of each fraction getscloser and closer to 1.Therefore, thelimit of the sequenceis:Example2:Limit by Direct SubstitutionEvaluate:
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Calculus - Limits - Page 2 preview imageStudy GuideThis expression is defined at (x = 2), so we can substitute directly:Example3:Limit Requiring FactoringEvaluate:Substituting (x =-3) gives:which isundefined. This means we must simplify first.Factor the numerator:Now simplify:Now take the limit:Graphical Interpretation
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Calculus - Limits - Page 3 preview imageStudy GuideFigure 1The graph of y = (x2−9)/(x + 3).The graph ofis the same as the graph of the line:except for one missing pointat ((-3,-6)).Even though the function is not defined at (x =-3), the limit still exists and equals–6.Example4:Simplifying a Compound FractionEvaluate:Substituting (x = 3) gives (0/0), so we simplify.
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Calculus - Limits - Page 4 preview imageStudy GuideAfter simplifying the compound fraction and canceling common factors, the expression reduces to:Now substitute:Example5:Limit That Equals ZeroEvaluate:Substitute (x = 0):Example6:Limit That Does Not ExistEvaluate:Substituting (x = 0) gives:which is undefined.The expression grows without bound as (x) approaches 0, so the limitdoes not exist (DNE).(Remember:infinity is not a real number, so it cannot be the value of a limit.)
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Calculus - Limits - Page 5 preview imageStudy GuideKey Takeaways•A limit describes what a valueapproaches, not necessarily what it equals•Trydirect substitutionfirst•If you get (0/0), simplify algebraically•A limit can exist even if the function isundefined at a point•Division by zero means the limitdoes not exist2.One-Sided Limits1. What Are One-Sided Limits?Sometimes, it is not enough to look at how a function behaves near a point fromboth sides.Instead, we may need to consider the behavior fromonly 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 existsonly if both one-sided limits exist and are equal.That is,if and only if
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Calculus - Limits - Page 6 preview imageStudy GuideIf the left-hand and right-hand limits are different—or if one does not exist—then the limitdoes notexist (DNE).Example1:Right-Hand LimitEvaluate:As (x) approaches 0from 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 isnot always true, as the nextexample shows.Example2:Left-Hand LimitEvaluate:As (x) approaches 0from the left, (x) is negative.The square root of a negative number isnot definedin the real numbers.So:Since:the two-sided limit
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Calculus - Limits - Page 7 preview imageStudy Guidealsodoes not exist.Example3:One-Sided Limits with Absolute ValueEvaluate the following limits:(a) Left-hand limitAs (x→2-), the expression (x-2) isnegative, so:Thus:So:(b) Right-hand limitAs (x→2+), (x-2) ispositive, so:Thus:
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Calculus - Limits - Page 8 preview imageStudy GuideSo:(c) Two-sided limitBecause:the two-sided limit:does not exist (DNE).Key Takeaways•One-sided limits look at behavior fromonly one direction•(x→c-) means from theleft•(x→c+) means from theright•A two-sided limit existsonly if both one-sided limits are equal•If left and right limits differ, the limit isDNE•Absolute value expressions often require checkingone-sided limits3.Infinite Limits1. What Is an Infinite Limit?Sometimes, as the input value (x) gets close to a certain number, a function doesnotapproach afinite value.
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Calculus - Limits - Page 9 preview imageStudy GuideInstead, 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 aninfinite limit.We write:2. Vertical AsymptotesIf a function has an infinite limit as (x) approaches a number (c), then the graph has averticalasymptoteat:This means the graph gets very close to the vertical line (x = c) but never touches it.3. Infinite Limits in Rational FunctionsArational functionoften has an infinite limit when:•The denominator approaches0•The numerator approaches anonzero valueThesignof the infinite limit depends on:•The sign of the numerator•The sign of the denominator near the point being approachedExample1:A Basic Infinite LimitEvaluate:
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Calculus - Limits - Page 10 preview imageStudy GuideAs (x) approaches 0:•The numerator (1) is always positive•The denominator (x^2) approaches 0 and is always positiveSo the fraction increases without bound:Figure 1The graph ofy= 1/x2.This figure shows that the graph rises sharply on both sides of (x = 0), confirming a vertical asymptoteat (x = 0).Example2:One-Sided Infinite LimitEvaluate:As (x) approaches 2from the left:•The numerator approaches (5) (positive)•The denominator approaches (0) throughnegative values
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Document Details

University
Texas A&M University
Course
Calculus
Subject
Mathematics

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