A limit asks what a function approaches, not what it equals. Try substitution first; if it gives 0/0, factor and cancel, or differentiate top and bottom (L'Hôpital). The graph shows the function near the point with the limit marked.
କାର୍ଯ୍ୟକାରୀ ଉଦାହରଣ · limit of sin(x)/x as x -> 0
ପଦକ୍ଷେପ କ୍ରମେ
- \lim_{x \to 0^+-} \frac{\sin{\left(x \right)}}{x}
Try direct substitution first.
- \frac{0}{0}
Substitution gives 0/0: an indeterminate form.
- \lim \frac{\cos{\left(x \right)}}{1}
L'Hôpital's rule: differentiate numerator and denominator separately.
- = 1
Substitute.
ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ
ଏବେ ଆପଣ ଗୋଟିଏ ସମସ୍ୟା ବାଛନ୍ତୁ, କିମ୍ବା ଆପଣଙ୍କର ନିଜ ସମସ୍ୟାକୁ ଟାଇପ କରନ୍ତୁ କିମ୍ବା ଆଙ୍କନ୍ତୁ। ପ୍ରତ୍ୟେକ ପର୍ଯ୍ୟାୟରେ, ଗୋଟିଏ ଚିତ୍ର, ଉତ୍ତର ଲୁକ୍କାୟିତ ହୋଇଥାଏ ଯେ ପର୍ଯ୍ୟନ୍ତ ଆପଣ ପଚାରନ୍ତି ନାହିଁ।
ଗୋଟିଏ ମୁକ୍ତ ଖାତା ପ୍ରତ୍ୟେକ ପାଠରେ ଟିପ୍ପଣୀ ଯୋଗ କରିଥାଏ, ଆପଣ ଯାହା ସମାପ୍ତ କରିଛନ୍ତି ତାହାର ଗୋଟିଏ ରେକର୍ଡ, ଆପଣଙ୍କର ସମାଧାନ ହୋଇଥିବା ସମସ୍ୟାଗୁଡ଼ିକ ଗୋଟିଏ ସ୍ଥାନରେ, ଏବଂ ଜଣେ ଶିକ୍ଷକ ଆପଣ ଏହି ପୃଷ୍ଠା ବିଷୟରେ ପଚାରିପାରିବେ। ଗଣିତ ନିଜେ ସମସ୍ତଙ୍କୁ ଖୋଲା ଅଛି, ଲଗଇନ ହୋଇଥିବା କିମ୍ବା ନଥିବା।
ସଦସ୍ୟତା ନିଅନ୍ତୁ ଲଗଇନଏଠାରେ ବ୍ୟବହୃତ ପ୍ରତୀକଗୁଡିକ
ସମ୍ପୂର୍ଣ୍ଣ ବର୍ଣ୍ଣନା ପାଇଁ ଯେକୌଣସି ପ୍ରତୀକକୁ ଟ୍ୟାପ କରନ୍ତୁ, ଗୋଟିଏ ଚିତ୍ର, ଏବଂ ଏଥିରେ ଥିବା ପ୍ରତ୍ୟେକ ଅକ୍ଷରର ଅର୍ଥ କଣ।
କିପରି: Limits
- Try direct substitution first.
- Substitution gives 0/0: an indeterminate form.
- L'Hôpital's rule: differentiate numerator and denominator separately.
- Substitute.
ଲୋକମାନେ ପଚାରୁଥିବା ପ୍ରଶ୍ନ
What is a derivative in one sentence?
The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.
What is an integral in one sentence?
The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.
Why are derivatives and integrals opposites?
That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.
When do I use substitution and when integration by parts?
Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.
ଅଧିକ Calculus
DerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests