Concept explainers
Continuity and limits with transcendental functions Determine the interval(s) on which the following functions are continuous; then analyze the given limits.
53.
Want to see the full answer?
Check out a sample textbook solutionChapter 2 Solutions
Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
Calculus & Its Applications (14th Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
University Calculus: Early Transcendentals (4th Edition)
- The limit lim x-> 0 e^x-1/x equals a derivative f' (c), for some function f(x) and some real number c. (i) find f(x) and c. (ii) Use the derivative of the function f(x) to evaluate the limit. (iii) Find an equaiton of the tangent line to y = f(x) at x = c, for the value of c you found in (i).arrow_forwardGuess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) Let f(x)= cos(9x)−cos(10x)/x2 We want to find the limit limx→0 cos(9x)−cos(10x)/x2Start by calculating the values of the function for the inputs listed in this table. x x f(x)f(x) 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 Based on the values in this table, it appears limx→0 cos(9x)−cos(10x)/x2=?arrow_forwardThe continuous extension of (sin x)x [0, Pi]a) Graph f(x)= (sinx)x on the interval 0 ≤ x ≤ Pi. What value would youassign to f(x) to make it continuous at x = 0?b) Verify your conclusion in part (a) by finding lim ?→0+ ?(?) with L’Hôpital’s Rule.arrow_forward
- Guess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) Let f(x)=cos(9x)-cos(5x)/x2 We want to find the limit limx→0 cos(9x)−cos(5x)/x2Start by calculating the values of the function for the inputs listed in this table. x x f(x)f(x) 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 Based on the values in this table, it appears limx→0 cos(9x)−cos(5x)/ x2arrow_forwardh(x) = { tan^-1 (x^2+2x+1), if x is less than or equal to -1 [[x + 1]], if -1 < x < 1 (note: [[x + 1]] is a greatest integer function) coth^-1 (x), if x is greater than or equal to 1 Determine if h is continuous at x = −1, 0, 1. If not, identify if it is removable, jump essential, or infinite essential. --- hi tutor, if you're going to use acronyms or terms on the solution or explanation please indicate the expanded acronym so I can fully understand :) thank uarrow_forwardlim xto1 of the function e(2^x) /(x^2) .. use L'Hoptial's Rulearrow_forward
- Guess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) Let f(x)=cos(9x)−cos(2x)/x^2 We want to find the limit limx→0 cos(9x)−cos(2x)/x^2 Start by calculating the values of the function for the inputs listed in this table. x f(x) 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 Based on the values in this table, it appears limx→0 cos(9x)−cos(2x)/x^2=arrow_forwardThe continuous extension of (sin x)^x [0,π] a) Graph f(x)= (sin x)^x on the interval 0 ≤ x ≤ π. What value would you assign to f(x) to make it continuous at x = 0? b) Verify your conclusion in part (a) by finding limx→0+ f(x) with L’Hôpital’s Rule. c) Returning to the graph, estimate the maximum value of f(x) on [0, π].About where is max f(x) taken on? d) Sharpen your estimate in part (c) by graphing ƒ in the same window to see where its graph crosses the x-axis. To simplify your work, you might want to delete the exponential factor from the expression for f' and graph just the factor that has a zero.arrow_forwardQ2)valuate the limit: Limit x approach to 2 [sin(pi.x)]/[x^2-x-2]arrow_forward
- Prove that Eulers constant e=(lim x->infinity symbol) [1+1/x]^xarrow_forwardPlanck’s Law says that the energy emitted by a black body is given by ((1/x5)/ (e(1/x) − 1)) where x is the wavelength. Use L’Hoptial’s Rule to compute the energy as the wavelength goes to 0. You may assume that this limit is indeterminate, so that L’Hopital’s Rule applies.arrow_forwardConsider the function g(x) = cos(1/x). We will investigate the limit behavior at x = 0. c) If n is an arbitrary positive integer, find points x1 and x2 (int terms of n) in the interval(-1/n, 1/n( such that g(x2) = 1 and g(x2) = -1. d) Explain (in a brief paragraph) why (c) implies that g does not have a limit at x = 0. e) Where is g continuous? Justify your answer. You may use facts from the textbook in Section 2.2 - 2.5.arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning