Limits of some transcendental functions
- 1. PERFORMANCE TASK # 1 A. Complete the following tables of values to investigate lim 𝑥→1 𝑥2 − 2𝑥 + 4
- 2. PERFORMANCE TASK # 1 B.
- 3. PERFORMANCE TASK # 1 C.
- 4. MOST ESSENTIAL LEARNING COMPETENCIES
- 5. LIMITS OF SOME TRANSCENDENTAL FUNCTIONS AND SOME INDETERMINATE FORMS BASIC CALCULUS_Q3_WEEK 2
- 6. LIMITS OF EXPONENTIAL, LOGARITHMIC,AND TRIGONOMETRIC FUNCTIONS Real-world situations can be expressed in terms of functional relationships.These functional relationships are called mathematical models. In applications of calculus, it is quite important that one can generate these mathematical models. They sometimes use functions that you encountered in precalculus, like the exponential, logarithmic, and trigonometric functions.
- 8. EVALUATING LIMITS OF EXPONENTIAL FUNCTIONS First, we consider the natural exponential function f(x) = 𝑒𝑥, where e is called the Euler number, and has value 2.718281.... EXAMPLE 1: Evaluate the lim 𝑥→0 𝑒𝑥 Solution. We will construct the table of values for f(x) = 𝑒𝑥. We start by approaching the number 0 from the left or through the values less than but close to 0.
- 9. Intuitively, from the table above, lim 𝑥→0− 𝑒𝑥 = 1 . Now we consider approaching 0 from its right or through values greater than but close to 0. From the table, as the values of x get closer and closer to 0, the values of f(x) get closer and closer to 1. So, lim 𝑥→0+ 𝑒𝑥 = 1 Combining the two one-sided limits allows us to conclude that lim 𝑥→0 𝑒𝑥 = 1
- 10. We can use the graph of f(x) = 𝑒𝑥 to determine its limit as x approaches 0. The figure below is the graph of f(x) = 𝑒𝑥.
- 11. EVALUATING LIMITS OF LOGARITHMIC FUNCTIONS Now, consider the natural logarithmic function f(x) = ln 𝑥. Recall that ln x = loge x. Moreover, it is the inverse of the natural exponential function y = 𝑒𝑥 . EXAMPLE 2: Evaluate lim 𝑥→1 ln 𝑥 Solution. We will construct the table of values for f(x) = ln x. We first approach the number 1 from the left or through values less than but close to 1.
- 12. Intuitively, lim 𝑥→1− ln 𝑥 = 0 . Now we consider approaching 1 from its right or through values greater than but close to 1. Intuitively, lim 𝑥→1+ ln 𝑥 = 0 . As the values of x get closer and closer to 1, the values of f(x) get closer and closer to 0. In symbols, lim 𝑥→1 ln 𝑥 = 0
- 13. We now consider the common logarithmic function f(x) = log10 x. Recall that f(x) = log10 x = log x. EXAMPLE 3: Evaluate lim 𝑥→1 log 𝑥 Now we consider approaching 1 from its right or through values greater than but close to 1. As the values of x get closer and closer to 1, the values of f(x) get closer and closer to 0. In symbols, lim 𝑥→1 log 𝑥 = 0
- 14. Consider now the graphs of both the natural and common logarithmic functions. We can use the following graphs to determine their limits as x approaches 1. a. lim 𝑥→𝑒 ln 𝑥 b. lim 𝑥→10 l𝑜𝑔 𝑥 c. lim 𝑥→3 ln 𝑥 d. lim 𝑥→3 l𝑜𝑔 𝑥 e. lim 𝑥→0+ l𝑛 𝑥 f. lim 𝑥→0+ l𝑜𝑔 𝑥 1 1 1.09.. 0.47.. -∞ -∞
- 15. TRIGONOMETRIC FUNCTIONS As the values of x get closer and closer to 1, the values of f(x) get closer and closer to 0. In symbols, lim 𝑥→0 sin 𝑥 = 0
- 16. We can also find lim 𝑥→0 sin 𝑥 by using the graph of the sine function. Consider the graph of f(x) = sin x. a. lim 𝑥→ 𝜋 2 sin 𝑥 b. lim 𝑥→𝜋 sin 𝑥 c. lim 𝑥→ −𝜋 2 sin 𝑥 d. lim 𝑥→−𝜋 sin 𝑥 =1 =0 =-1 =0
- 17. SOME SPECIAL LIMITS We will determine the limits of three special functions; namely, f(t) = sin 𝑡 𝑡 , g(t) = 1−cos 𝑡 𝑡 , and h(t) = 𝑒𝑡−1 𝑡 These functions will be vital to the computation of the derivatives of the sine, cosine, and natural exponential functions.
- 18. THREE SPECIAL FUNCTIONS EXAMPLE 1: Evaluate lim 𝑥→0 sin 𝑡 𝑡 Solution. We will construct the table of values for f(t) = sin 𝑡 𝑡 . We first approach the number 0 from the left or through values less than but close to 0. Now we consider approaching 0 from the right or through values greater than but close to 0.
- 20. EXAMPLE 2: Evaluate lim 𝑥→0 1−cos 𝑡 𝑡 Solution. We will construct the table of values for g(t) = 1−cos 𝑡 𝑡 . We first approach the number 1 from the left or through the values less than but close to 0. Now we consider approaching 0 from the right or through values greater than but close to 0.
- 22. EXAMPLE 3: Evaluate lim 𝑥→0 𝑒𝑡−1 𝑡 Solution. We will construct the table of values for h(t) = 𝑒𝑡−1 𝑡 We first approach the number 0 from the left or through the values less than but close to 0. Now we consider approaching 0 from the right or through values greater than but close to 0.
- 24. INDETERMINATE FORM 𝟎 𝟎 There are functions whose limits cannot be determined immediately using the Limit Theorems we have so far. In these cases, the functions must be manipulated so that the limit, if it exists, can be calculated. We call such limit expressions indeterminate forms.
- 26. 2. Evaluate lim 𝑥→1 𝑥2−1 𝑥 −1
- 27. PERFORMANCE TASK # 2 𝐴. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑙𝑖𝑚𝑖𝑡𝑠 1. lim 𝑥→1 3𝑥 2. lim 𝑥→2 5𝑥 3. lim 𝑥→4 log 𝑥 4. lim 𝑥→0 cos 𝑥 5. lim 𝑥→0 tan 𝑥 6. lim 𝑡→𝝅 𝑡 sin 𝑡 8. lim 𝑡→−1 𝑡2 − 1 𝑡2 + 4t + 3 7. lim 𝑡→ 𝜋 2 cos 𝑡 sin t 9. lim 𝑥→−1 𝑥2 − 𝑥 − 2 𝑥3 + 6𝑥2 − 7x 10. lim 𝑥→16 𝑥2 − 256 4 − 𝑥