Limits Problems and Solutions

Limit methods

Direct substitution

Factorization

$(1).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,5}{\normalsize \dfrac{x^2-25}{x-5}}$

$(2).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,1}{\normalsize \dfrac{1-x^{-\dfrac{1}{3}}}{1-x^{-\dfrac{2}{3}}}}$

$(3).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^3+7x^2-36}{x^2+2x-8}}$

List of limits questions on factorisation to learn how to find the limit of a function by factoring.

Rationalization

$(1).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sqrt{1+x}-1}{x}}$

$(2).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,-1}{\normalsize \dfrac{\sqrt[\displaystyle 3]{7-x}-2}{x+1}}$

$(3).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{x}{\sqrt{a+x}-\sqrt{a-x}}}$

List of limit questions on rationalisation with solutions to learn how to find the limits by rationalization.

L’Hôpital’s Rule

d

Limit of functions

Algebraic functions

List of limit problems with solutions for the algebraic functions to find the limits of functions in which algebraic functions are involved.

Trigonometric functions

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x-\sin{x}}{x^3}}$

List of limit problems with solutions for the trigonometric functions to find the limits of functions in which trigonometric functions are involved.

Logarithmic functions

List of limit problems with solutions for the logarithmic functions to find the limits of functions in which logarithmic functions are involved.

Exponential functions

List of limit problems with solutions for the exponential functions to evaluate the limits of functions in which exponential functions are involved.

Difficult Problems

List of hard/tough problems in limits with solutions.

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